Similar figures have identical shapes but different sizes, characterized by congruent corresponding angles and proportional corresponding sides.
If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar: $\triangle ABC \sim \triangle DEF$
Corresponding angles are equal: $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$
Ratio of corresponding sides is equal to the scale factor $k$: $\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC} = k$
📘 Traditional Algebraic Method
Identify matching angle pairs using parallel line properties or shared vertices, write out the correct similarity statement vertex-by-vertex, and set up ratios of corresponding sides.
⚡ SAT Speed Trick & Desmos Hack
Use the Desmos geometry overlay or set up proportion equations directly in the calculator input line as $x/a = y/b$ to solve instantly.
💡 Worked SAT Archetype Example
Problem: In triangles $\triangle ABC$ and $\triangle XYZ$, $\angle A = \angle X$ and $\angle B = \angle Y$. If $AB = 6$, $BC = 8$, and $XY = 9$, what is the length of side $YZ$?
📘 Step-by-Step Textbook Solution:
Step 1
Identify that $\triangle ABC \sim \triangle XYZ$ by the AA Similarity Postulate.
Step 2
Set up the ratio of corresponding sides: $\frac{YZ}{BC} = \frac{XY}{AB}$
Step 3
Substitute the known values into the proportion equation: $\frac{YZ}{8} = \frac{9}{6}$
Triangles are similar if an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides including these angles are proportional (SAS), or if all three pairs of corresponding sides are proportional (SSS).
SAS Similarity: $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A = \angle D \implies \triangle ABC \sim \triangle DEF$
Proportionality constant $k$ applies uniformly across all linear dimensions.
📘 Traditional Algebraic Method
Verify that all three side ratios match or that two side ratios match alongside the included angle, then solve the resulting linear equation for the unknown variable.
⚡ SAT Speed Trick & Desmos Hack
Define variables in Desmos as functions or equations and solve the proportional equality using numerical solvers.
💡 Worked SAT Archetype Example
Problem: Triangle $ABC$ has side lengths $AB = 5$, $BC = 7$, and $AC = 10$. Triangle $DEF$ has side lengths $DE = 15$, $EF = 21$, and $DF = x$. If $\triangle ABC \sim \triangle DEF$, what is the value of $x$?
📘 Step-by-Step Textbook Solution:
Step 1
Identify corresponding sides based on given side lengths: $\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}$
Step 2
Calculate the scale factor using the smallest sides: $\frac{15}{5} = 3$
A line parallel to one side of a triangle divides the other two sides proportionally, creating a smaller similar triangle nested inside or adjacent to the original triangle.
Triangle Proportionality Theorem: If $DE \parallel BC$, then $\frac{AD}{DB} = \frac{AE}{EC}$
Corollary ratio of parts to wholes: $\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$
Split the nested figure into two separate triangles if needed, define unknown segment lengths using variable expressions like $x$ or total minus part, and solve the rational equation.
⚡ SAT Speed Trick & Desmos Hack
Set up the rational equation directly in Desmos and check intersection with the domain constraints.
💡 Worked SAT Archetype Example
Problem: In $\triangle ABC$, point $D$ is on $AB$ and point $E$ is on $AC$ such that $DE \parallel BC$. If $AD = 4$, $DB = 6$, and $AE = 5$, what is the length of $EC$?
📘 Step-by-Step Textbook Solution:
Step 1
Apply the Triangle Proportionality Theorem: $\frac{AD}{DB} = \frac{AE}{EC}$
Step 2
Substitute the known values: $\frac{4}{6} = \frac{5}{EC}$
Step 3
Cross-multiply to solve for $EC$: $4 \times EC = 30$
Step 4
Divide by 4: $EC = 7.5$
⚡ Speed / Desmos Tactic:
Step 1
Type the proportion into Desmos: $4/6 = 5/x$
Step 2
Locate the solution line for $x$: $x = 7.5$
Concept 4
Concept 4: Area and Volume Ratios of Similar Figures
When two geometric figures are similar with a linear scale factor of $k$, their perimeter scales by $k$, their area scales by $k^2$, and their volume scales by $k^3$.
Linear scale factor (sides, heights, perimeters, radii): ratio = $k$
Area scale factor (surface area, cross-sectional area): ratio = $k^2$
Volume scale factor (3D capacity, displacement): ratio = $k^3$
📘 Traditional Algebraic Method
Determine the linear scale factor $k$ first, square it to find the area ratio or cube it to find the volume ratio, and multiply the given measurement by this scaling factor.
⚡ SAT Speed Trick & Desmos Hack
Store $k$ as a variable in Desmos (e.g., $k = ext{new}/ ext{old}$) and calculate $A_{new} = A_{old} \cdot k^2$ instantly.
💡 Worked SAT Archetype Example
Problem: Two similar cylinders have heights in the ratio of $2:3$. If the volume of the smaller cylinder is $16\text{ cm}^3$, what is the volume of the larger cylinder?
📘 Step-by-Step Textbook Solution:
Step 1
Identify the linear scale factor $k$: $k = \frac{3}{2}$
Step 2
Calculate the volume scale factor by cubing $k$: $k^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$
Step 3
Set up the volume relationship: $V_{larger} = V_{smaller} \times k^3$
Step 4
Compute the final volume: $V_{larger} = 16 \times \frac{27}{8} = 54$
⚡ Speed / Desmos Tactic:
Step 1
Enter into Desmos: $16 \times (3/2)^3$
Step 2
Read output: $54$
Concept 5
Concept 5: Right Triangle Similarity & Geometric Mean
The altitude drawn to the hypotenuse of a right triangle creates three mutually similar right triangles, leading to the geometric mean relationships.
Altitude rule: $h^2 = x \cdot y$, where $h$ is the altitude to the hypotenuse and $x, y$ are the segments of the hypotenuse
Leg rule: $\text{leg}^2 = \text{nearest segment} \times \text{hypotenuse}
All three triangles ($\triangle ABC$, $\triangle DBA$, $\triangle DAC$) are mutually similar
📘 Traditional Algebraic Method
Sketch the separate triangles with aligned vertices to confirm corresponding sides, then apply proportions or geometric mean formulas.
⚡ SAT Speed Trick & Desmos Hack
Memorize the geometric mean formulas and evaluate directly using calculator square root functions.
💡 Worked SAT Archetype Example
Problem: In right triangle $ABC$ with right angle at $C$, altitude $CD$ is drawn to hypotenuse $AB$. If $AD = 4$ and $DB = 9$, what is the length of altitude $CD$?
📘 Step-by-Step Textbook Solution:
Step 1
Identify the altitude rule for right triangles: $CD^2 = AD \times DB$
Step 2
Substitute the given segment lengths: $CD^2 = 4 \times 9$
Step 3
Simplify the product: $CD^2 = 36$
Step 4
Take the positive square root: $CD = 6$
⚡ Speed / Desmos Tactic:
Step 1
Input into Desmos: $\sqrt{4 \times 9}$
Step 2
Read output: $6$
Practice Questions (20)
Question 1Scale Factor and Side Lengths
Easy
Triangle $ABC$ is similar to triangle $DEF$. If $AB = 6$, $BC = 8$, and $DE = 12$, what is the length of $EF$?
Hint: The ratio of corresponding sides in similar triangles is constant.
📘 Step-by-Step Algebraic Solution
Step 1
Identify the scale factor $k = DE / AB = 12 / 6 = 2$.
Step 2
Set up the ratio $EF / BC = k$.
Step 3
$EF / 8 = 2$.
Step 4
$EF = 16$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Note that $DE$ is double $AB$.
Step 2
Double $BC$ to get $EF$.
Step 3
$8 \times 2 = 16$.
Question 2Scale Factor and Side Lengths
Easy
Two similar rectangles have areas of $25$ and $100$. If the width of the smaller rectangle is $5$, what is the width of the larger rectangle?
Hint: The ratio of areas is the square of the scale factor.
📘 Step-by-Step Algebraic Solution
Step 1
Ratio of areas is $100 / 25 = 4$.
Step 2
The scale factor $k$ is $\sqrt{4} = 2$.
Step 3
Multiply the smaller width by $k$: $5 \times 2 = 10$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Area ratio is $4:1$.
Step 2
Linear ratio is $\sqrt{4}:\sqrt{1} = 2:1$.
Step 3
$5 \times 2 = 10$.
Question 3Scale Factor and Side Lengths
Medium
Triangle $PQR$ is similar to triangle $STU$. The perimeter of $PQR$ is $40$ and the perimeter of $STU$ is $60$. If $PQ = 10$, what is the length of $ST$?
Hint: The ratio of perimeters is equal to the ratio of corresponding sides.
📘 Step-by-Step Algebraic Solution
Step 1
Ratio of perimeters is $60 / 40 = 1.5$.
Step 2
$ST / PQ = 1.5$.
Step 3
$ST / 10 = 1.5$.
Step 4
$ST = 15$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Perimeter ratio is $6/4 = 3/2$.
Step 2
Multiply $PQ$ by $3/2$.
Step 3
$10 \times 1.5 = 15$.
Question 4Scale Factor and Side Lengths
Medium
A triangle with sides $3, 4, 5$ is similar to a triangle with a perimeter of $36$. What is the length of the longest side of the larger triangle?
Hint: Find the perimeter of the smaller triangle first.
📘 Step-by-Step Algebraic Solution
Step 1
Perimeter of small triangle $= 3 + 4 + 5 = 12$.
Step 2
Scale factor $k = 36 / 12 = 3$.
Step 3
Longest side of small triangle is $5$.
Step 4
$5 \times 3 = 15$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$36 / (3+4+5) = 3$.
Step 2
$5 \times 3 = 15$.
Question 5Scale Factor and Side Lengths
Hard
Two similar triangles have areas in ratio $9:16$. If the sum of the lengths of the corresponding sides is $35$, what is the length of the longer side?
Hint: The ratio of sides is the square root of the ratio of areas.
📘 Step-by-Step Algebraic Solution
Step 1
Ratio of sides $k_1 : k_2 = \sqrt{9} : \sqrt{16} = 3 : 4$.
Step 2
Let sides be $3x$ and $4x$.
Step 3
$3x + 4x = 35 \implies 7x = 35 \implies x = 5$.
Step 4
Longer side is $4x = 4(5) = 20$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Ratio is $3:4$.
Step 2
$35 \times (4 / (3+4)) = 35 \times (4/7) = 20$.
Question 6Parallel Lines and Intercepts
Easy
In $\triangle ABC$, $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$. If $AD = 2$, $DB = 3$, and $AE = 4$, what is $EC$?
Hint: Use the Triangle Proportionality Theorem: $AD/DB = AE/EC$.
📘 Step-by-Step Algebraic Solution
Step 1
$AD / DB = AE / EC$.
Step 2
$2 / 3 = 4 / EC$.
Step 3
$2 \times EC = 12$.
Step 4
$EC = 6$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$AD$ is $2$, $AE$ is $4$ (doubled).
Step 2
$DB$ is $3$, so $EC$ must be $3 \times 2 = 6$.
Question 7Parallel Lines and Intercepts
Easy
In $\triangle ABC$, $DE \parallel BC$. If $AD = 3$ and $AB = 9$, what is the ratio of $DE$ to $BC$?
Hint: The ratio of the small triangle side to the large triangle side is $AD/AB$.
📘 Step-by-Step Algebraic Solution
Step 1
$\triangle ADE \sim \triangle ABC$.
Step 2
Ratio $= AD / AB = 3 / 9$.
Step 3
Simplify $3/9 = 1/3$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$3/9 = 1/3$.
Question 8Parallel Lines and Intercepts
Medium
In $\triangle ABC$, $DE \parallel BC$. If $AD = 4$, $DB = 2$, and $BC = 9$, what is $DE$?
Hint: The ratio $AD/AB = DE/BC$. Note $AB = AD + DB$.
📘 Step-by-Step Algebraic Solution
Step 1
$AB = 4 + 2 = 6$.
Step 2
$AD / AB = DE / BC \implies 4 / 6 = DE / 9$.
Step 3
$2 / 3 = DE / 9$.
Step 4
$DE = 6$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$DE = BC \times (AD / (AD+DB))$.
Step 2
$9 \times (4/6) = 6$.
Question 9Parallel Lines and Intercepts
Medium
A line parallel to the base of a triangle divides the other two sides into segments of lengths $2$ and $5$ (starting from the vertex). What is the ratio of the area of the smaller triangle to the area of the larger triangle?
Hint: The ratio of areas is the square of the ratio of corresponding sides.
📘 Step-by-Step Algebraic Solution
Step 1
Small side $= 2$.
Step 2
Large side $= 2 + 5 = 7$.
Step 3
Ratio of sides $= 2/7$.
Step 4
Ratio of areas $= (2/7)^2 = 4/49$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$(2 / (2+5))^2 = (2/7)^2 = 4/49$.
Question 10Parallel Lines and Intercepts
Hard
In $\triangle ABC$, $DE \parallel BC$. The area of $\triangle ADE$ is $16$ and the area of trapezoid $DBCE$ is $20$. What is the ratio $AD:DB$?
Hint: Area of $\triangle ABC = Area(\triangle ADE) + Area(Trapezoid)$.
📘 Step-by-Step Algebraic Solution
Step 1
Area $\triangle ABC = 16 + 20 = 36$.
Step 2
Ratio of areas $\triangle ADE / \triangle ABC = 16 / 36 = 4 / 9$.
Question 11Similar Triangles in Circles/Coordinates
Easy
In the coordinate plane, $\triangle ABC$ has vertices $(0,0), (3,0), (0,4)$. $\triangle DEF$ is similar to $\triangle ABC$ with a scale factor of $2$. What is the area of $\triangle DEF$?
Hint: Area of $\triangle ABC = 0.5 \times 3 \times 4 = 6$. Multiply by $k^2$.
📘 Step-by-Step Algebraic Solution
Step 1
Area $\triangle ABC = 0.5 \times 3 \times 4 = 6$.
Step 2
Scale factor $k = 2$.
Step 3
Area $\triangle DEF = Area(\triangle ABC) \times k^2$.
Step 4
$6 \times 2^2 = 6 \times 4 = 24$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$0.5 \times 3 \times 4 = 6$.
Step 2
$6 \times 4 = 24$.
Question 12Similar Triangles in Circles/Coordinates
Easy
Two chords $AB$ and $CD$ intersect at point $P$ inside a circle. If $AP = 4, PB = 3, CP = 2$, what is $PD$?
Question 13Similar Triangles in Circles/Coordinates
Medium
A triangle has vertices $(0,0), (6,0), (0,8)$. A similar triangle has vertices $(0,0), (3,0), (0,y)$. What is $y$?
Hint: The ratio of legs must be consistent.
📘 Step-by-Step Algebraic Solution
Step 1
Legs of first triangle are $6$ and $8$.
Step 2
Legs of second triangle are $3$ and $y$.
Step 3
$3 / 6 = y / 8$.
Step 4
$0.5 = y / 8 \implies y = 4$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$6$ became $3$ (divided by $2$).
Step 2
$8$ divided by $2$ is $4$.
Question 14Similar Triangles in Circles/Coordinates
Medium
In a circle, a secant from point $P$ intersects the circle at $A$ and $B$. If $PA = 3$ and $PB = 12$, what is the length of a tangent segment $PT$ from $P$ to the circle?
Hint: Tangent-Secant Theorem: $PT^2 = PA \times PB$.
📘 Step-by-Step Algebraic Solution
Step 1
$PT^2 = PA \times PB$.
Step 2
$PT^2 = 3 \times 12 = 36$.
Step 3
$PT = \sqrt{36} = 6$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$\sqrt{3 \times 12} = 6$.
Question 15Similar Triangles in Circles/Coordinates
Hard
In $\triangle ABC$, $\angle A = 90^\circ$ and $AD$ is the altitude to $BC$. If $BD = 4$ and $DC = 9$, what is $AD$?
Hint: Geometric Mean Theorem: $AD^2 = BD \times DC$.
📘 Step-by-Step Algebraic Solution
Step 1
$\triangle ABD \sim \triangle CAD$.
Step 2
$AD / BD = DC / AD$.
Step 3
$AD^2 = 4 \times 9 = 36$.
Step 4
$AD = 6$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$\sqrt{4 \times 9} = 6$.
Question 16Right Triangle Similarity
Easy
A right triangle has legs $6$ and $8$. An altitude is drawn to the hypotenuse. What is the length of the hypotenuse?
Hint: Use the Pythagorean theorem.
📘 Step-by-Step Algebraic Solution
Step 1
$a^2 + b^2 = c^2$.
Step 2
$6^2 + 8^2 = c^2$.
Step 3
$36 + 64 = 100 = c^2$.
Step 4
$c = 10$.
⚡ Desmos Shortcut / Speed Hack
Step 1
Recognize $3-4-5$ triangle scaled by $2$.
Question 17Right Triangle Similarity
Easy
In a right triangle, the altitude to the hypotenuse divides it into segments of $2$ and $8$. What is the length of the altitude?
Hint: Geometric Mean Theorem: $h^2 = p \times q$.
📘 Step-by-Step Algebraic Solution
Step 1
$h^2 = 2 \times 8$.
Step 2
$h^2 = 16$.
Step 3
$h = 4$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$\sqrt{2 \times 8} = 4$.
Question 18Right Triangle Similarity
Medium
In a right triangle with legs $5$ and $12$, what is the length of the altitude to the hypotenuse?
In a right triangle, the altitude to the hypotenuse has length $6$. One segment of the hypotenuse is $4$. What is the length of the other segment?
Hint: $h^2 = p \times q$.
📘 Step-by-Step Algebraic Solution
Step 1
$6^2 = 4 \times q$.
Step 2
$36 = 4q$.
Step 3
$q = 9$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$36 / 4 = 9$.
Question 20Right Triangle Similarity
Hard
In $\triangle ABC$, $\angle C = 90^\circ$. $CD$ is the altitude to $AB$. If $AC = 6$ and $AD = 3.6$, what is $AB$?
Hint: Leg Rule: $AC^2 = AD \times AB$.
📘 Step-by-Step Algebraic Solution
Step 1
$AC^2 = AD \times AB$.
Step 2
$6^2 = 3.6 \times AB$.
Step 3
$36 = 3.6 \times AB$.
Step 4
$AB = 36 / 3.6 = 10$.
⚡ Desmos Shortcut / Speed Hack
Step 1
$36 / 3.6 = 10$.
Official SAT Exam Blueprint & Weightage
Metric
Exam Pattern & Weightage
Question Frequency
1-3 questions per test module
Module 1 Appearance
Medium Frequency (Foundational tests)
Module 2 Appearance
High Frequency (Hard module score differentiator)
Pattern Analysis
College Board heavily tests similarity combined with coordinate geometry (scaling from the origin) and nested triangles cut by transversals. Module 2 hard questions frequently test area-to-perimeter scaling ratios and indirect measurement word problems.
🏛️
Official SAT PYQ Drill Bank (2023–2026)
We are compiling real verified College Board exam patterns with step-by-step Desmos shortcuts for Similarity.
Updated for 2026 Testing Season
Revision, Traps & Desmos Syntax
Side Proportionality
$\frac{A'B'}{AB} = k$
Ratio of corresponding sides equals scale factor $k$.
Area Scaling Ratio
$\frac{\text{Area}_2}{\text{Area}_1} = k^2$
Area scales by the square of the linear scale factor.
Volume Scaling Ratio
$\frac{\text{Volume}_2}{\text{Volume}_1} = k^3$
Volume scales by the cube of the linear scale factor.
Geometric Mean (Altitude)
$h = \sqrt{x y}$
Altitude to hypotenuse equals geometric mean of hypotenuse segments.
🚨 Top SAT Traps & Misconceptions
⚠️ SAT Trap: Confusing Linear and Area Ratios
Applying the linear scale factor $k$ directly to area or volume questions instead of squaring ($k^2$) or cubing ($k^3$). Always check unit dimensions.
⚠️ SAT Trap: Incorrect Vertex Matching
Assuming alphabetical order represents corresponding vertices in similarity statements. Always match vertices by their congruent angle markings.
⚡ Essential Desmos Cheatsheet
🎯 Proportion Solver
a/b = c/x
Type proportions directly with an unknown variable $x$ to let Desmos solve linear and rational equations instantly.
🎯 Scaling Factor Storage
k = new_side / old_side
Store the scale factor as $k$ and compute scaled areas using $k^2$ without manual recalculation.
3-Level Mock Test (30 Questions)
🟢 Level 1: Foundation
10 Qs · Sub-600 Score
🟡 Level 2: Target 700+
10 Qs · 600–740 Score
🔴 Level 3: 800-Mastery
10 Qs · 750–800 Score
Question 1Level 1: Foundation
Triangle $ABC$ is similar to triangle $DEF$. If $AB = 4$ and $DE = 8$, what is the scale factor from triangle $ABC$ to triangle $DEF$?
Explanation:
Step 1
Identify the ratio of corresponding sides.
Step 2
$k = \frac{DE}{AB} = \frac{8}{4}$
Step 3
$k = 2$
Question 2Level 1: Foundation
Two triangles are similar. If the sides of the first triangle are 3, 4, and 5, and the shortest side of the second triangle is 9, what is the length of the longest side of the second triangle?
Explanation:
Step 1
Determine the scale factor by comparing the shortest sides.
Step 2
$k = \frac{9}{3} = 3$
Step 3
Multiply the longest side of the first triangle by the scale factor: $5 \times 3 = 15$
Question 3Level 1: Foundation
In $\triangle ABC$, $DE \parallel BC$ where $D$ is on $AB$ and $E$ is on $AC$. If $AD=2, DB=2, AE=3$, what is $EC$?
If two triangles are similar, which of the following must be true?
Explanation:
Step 1
Recall the definition of similar triangles.
Step 2
Similar triangles have congruent corresponding angles and proportional side lengths.
Step 3
Option C is the definition of similarity.
Question 5Level 1: Foundation
A triangle has sides 5, 12, 13. A similar triangle has a perimeter of 60. What is the length of the shortest side of the second triangle?
Explanation:
Step 1
Find the perimeter of the first triangle: $5+12+13 = 30$.
Step 2
Find the scale factor: $k = \frac{60}{30} = 2$.
Step 3
Multiply the shortest side by the scale factor: $5 \times 2 = 10$.
Question 6Level 1: Foundation
If $\triangle XYZ \sim \triangle PQR$, which angle is congruent to $\angle Y$?
Explanation:
Step 1
Identify corresponding vertices in the similarity statement.
Step 2
$X \leftrightarrow P, Y \leftrightarrow Q, Z \leftrightarrow R$.
Step 3
Therefore, $\angle Y \cong \angle Q$.
Question 7Level 1: Foundation
A 5-foot pole casts a 2-foot shadow. At the same time, a nearby tree casts a 10-foot shadow. How tall is the tree?
Explanation:
Step 1
Set up a proportion: $\frac{\text{height}}{\text{shadow}} = \frac{5}{2}$.
Step 2
$\frac{h}{10} = \frac{5}{2}$
Step 3
$h = 25$ feet.
Question 8Level 1: Foundation
If the ratio of the areas of two similar triangles is 1:4, what is the ratio of their corresponding sides?
Explanation:
Step 1
Recall that the ratio of areas is the square of the ratio of sides.
Step 2
$k^2 = \frac{1}{4}$
Step 3
$k = \sqrt{\frac{1}{4}} = \frac{1}{2}$.
Question 9Level 1: Foundation
In $\triangle ABC$, $\angle A = 40^\circ$ and $\angle B = 60^\circ$. In $\triangle DEF$, $\angle D = 40^\circ$ and $\angle E = 60^\circ$. Are the triangles similar?
Explanation:
Step 1
Check the Angle-Angle (AA) similarity criterion.
Step 2
Two angles are congruent, so the third must also be congruent.
Step 3
The triangles are similar by AA.
Question 10Level 1: Foundation
A rectangle has dimensions 4 by 6. A similar rectangle has a width of 8. What is its length?
Explanation:
Step 1
Find the scale factor: $k = \frac{8}{4} = 2$.
Step 2
Multiply the other dimension by the scale factor: $6 \times 2 = 12$.
Step 3
The length is 12.
Question 1Level 2: Target 700+
In $\triangle ABC$, $D$ is on $AB$ and $E$ is on $AC$ such that $\angle ADE = \angle ACB$. If $AD=3, AE=4, DB=1$, what is $EC$?
Explanation:
Step 1
Identify similar triangles $\triangle ADE \sim \triangle ACB$.