Similarity, Proof, and Trigonometry
Unit 2 of Geometry. 14 questions below, each with the working. Every answer was checked by a second pass before it was published.
Dilations and similarity, similarity criteria, proving theorems with similarity, right-triangle trigonometry, the Pythagorean theorem, applications.
How this unit is tested
Start by sorting any problem into one of three buckets: transformation/similarity, proof, or right-triangle computation. If the problem gives coordinates or asks you to compare figure sizes, it's a dilation question — find the scale factor first, since almost everything else follows from it. If the problem gives two triangles and asks whether they're similar, check angles first (AA is the fastest test); only fall back to SAS or SSS similarity when side lengths are given and angles aren't obviously matching.
For proof problems, look for a shared angle (reflexive property), vertical angles, or a pair of parallel lines cut by a transversal — these are the usual sources of the congruent angles that let you invoke AA. Many 'prove the theorem' problems (side-splitter, geometric mean relationships, midsegment) are really AA similarity proofs in disguise: draw the auxiliary line if needed, name the two triangles, match up angles, then write the proportion.
For right-triangle computation, always redraw the triangle and label the reference angle, the side opposite it, the side adjacent to it, and the hypotenuse before choosing sine, cosine, or tangent. If you're given two sides and need an angle, use the inverse trig function; if you're given an angle and one side, multiply or divide by the appropriate ratio. Reserve the Pythagorean theorem for when you have (or want) two legs and a hypotenuse and no angle is involved.
For applied problems (shadows, ladders, angles of elevation/depression), the whole task is translating the word problem into a labeled right triangle or a pair of similar triangles — once that diagram exists, the computation is routine. Do that translation step explicitly, even under time pressure, because that's where marks are actually lost.
What you have to know
Dilation
A dilation centered at point P with scale factor k maps a point A to A′ on ray PA such that PA′ = k·PA. Dilations preserve angle measure and orientation of the figure but scale all lengths by k; the image is similar to the original figure.
AA Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
SAS and SSS Similarity Theorems
Two triangles are similar if two pairs of corresponding sides are proportional and the included angles are congruent (SAS), or if all three pairs of corresponding sides are proportional (SSS).
Side-Splitter (Triangle Proportionality) Theorem
If a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally: if $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$, then $\dfrac{AD}{DB}=\dfrac{AE}{EC}$.
Right Triangle Altitude (Geometric Mean) Relationships
If the altitude is drawn from the right angle to the hypotenuse of a right triangle, it creates two smaller triangles similar to each other and to the original triangle. The altitude is the geometric mean of the two hypotenuse segments, and each leg is the geometric mean of the hypotenuse and the segment adjacent to that leg.
Pythagorean Theorem and Converse
In a right triangle with legs $a,b$ and hypotenuse $c$: $a^2+b^2=c^2$. Conversely, if a triangle's side lengths satisfy $a^2+b^2=c^2$ with $c$ the longest side, the triangle is a right triangle.
14 practice questions
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Triangle ABC has vertices A(2,3), B(4,3), C(4,7). It is dilated by scale factor 2 centered at the origin. Find the coordinates of A′.
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Answer. A′ = (4, 6)
A dilation centered at the origin multiplies each coordinate by the scale factor: $A' = (2\cdot2,\ 2\cdot3) = (4,6)$. -
A triangle with a side of length 20 is dilated by scale factor 3/4. What is the length of the corresponding side in the image?
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Answer. 15
Image length equals scale factor times original length: $\dfrac{3}{4}\times 20 = 15$. -
Triangle ABC has angle A = 48° and angle B = 82°. Triangle XYZ has angle X = 48° and angle Z = 50°. Are the two triangles similar?
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Answer. Yes, similar by AA
Angle C in triangle ABC is $180-48-82=50°$, and angle Y in triangle XYZ is $180-48-50=82°$. Both triangles have angle measures 48°, 82°, 50°, giving two (in fact three) pairs of congruent angles, so AA guarantees similarity. -
Triangle PQR has PQ = 8, PR = 12, angle P = 40°. Triangle STU has ST = 6, SU = 9, angle S = 40°. Are the triangles similar, and if so by what criterion?
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Answer. Yes, similar by SAS, with scale factor 3/4 from PQR to STU
The included angles are congruent (both 40°) and the sides forming those angles are proportional: $\dfrac{6}{8}=\dfrac{9}{12}=\dfrac{3}{4}$, satisfying the SAS Similarity Theorem. -
Triangle ABC has sides 6, 8, 10. Triangle DEF has sides 9, 12, 15. Are the triangles similar? Give the scale factor from ABC to DEF.
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Answer. Yes, similar by SSS; scale factor 3/2
All corresponding sides are proportional: $\dfrac{9}{6}=\dfrac{12}{8}=\dfrac{15}{10}=1.5$, satisfying SSS similarity. -
A person 5 feet tall casts an 8-foot shadow. At the same time, a flagpole casts a 24-foot shadow. How tall is the flagpole?
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Answer. 15 feet
The two right triangles are similar by AA (right angle plus equal angle of elevation to the sun), so $\dfrac{h}{24}=\dfrac{5}{8}$, giving $h=15$ feet. -
In triangle ABC, segment DE is parallel to BC, with D on AB and E on AC. AD = 4, DB = 6, AE = x, EC = 9. Find x.
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Answer. x = 6
By the Side-Splitter Theorem, $\dfrac{AD}{DB}=\dfrac{AE}{EC}$, so $\dfrac{4}{6}=\dfrac{x}{9}$, giving $x=6$. -
In right triangle ABC with the right angle at C, altitude CD is drawn to hypotenuse AB, splitting it into segments AD = 4 and DB = 9. Find the length of the altitude CD.
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Answer. CD = 6
The altitude to the hypotenuse is the geometric mean of the two segments it creates: $CD=\sqrt{AD\cdot DB}=\sqrt{4\cdot9}=\sqrt{36}=6$. -
When the altitude from the right angle is drawn to the hypotenuse of a right triangle, three triangles are formed. Which similarity criterion proves all three triangles similar to each other?
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Answer. AA similarity
Each of the two smaller triangles shares one acute angle with the original triangle and has a right angle from the altitude, giving two pairs of congruent angles in every comparison — enough for AA. -
In a right triangle, one acute angle measures 30° and the hypotenuse is 10. Find the length of the side opposite the 30° angle.
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Answer. 5
Using sine: $\sin(30^\circ)=\dfrac{\text{opposite}}{10}$, so opposite $=10\sin(30^\circ)=10\times0.5=5$. -
A right triangle has legs of length 5 and 12, and hypotenuse 13. Find the measure of the angle opposite the leg of length 5, to the nearest tenth of a degree.
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Answer. About 22.6°
Using inverse sine: $\sin^{-1}\left(\dfrac{5}{13}\right)\approx 22.6^\circ$, since the opposite leg is 5 and the hypotenuse is 13. -
A right triangle has legs of length 9 and 12. Find the length of the hypotenuse.
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Answer. 15
By the Pythagorean theorem, $c=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$. -
A triangle has side lengths 7, 24, and 25. Is it a right triangle?
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Answer. Yes
Checking the converse of the Pythagorean theorem with the two shorter sides: $7^2+24^2=49+576=625=25^2$, so the triangle is a right triangle. -
From a point on the ground 100 feet from the base of a building, the angle of elevation to the top of the building is 35°. How tall is the building, to the nearest foot?
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Answer. About 70 feet
The horizontal distance and height form a right triangle with the 35° angle at the observer's eye, so $\tan(35^\circ)=\dfrac{h}{100}$, giving $h=100\tan(35^\circ)\approx70$ feet.
What people get wrong
- Writing a similarity proportion with mismatched vertices (e.g., $AB/DE$ when the similarity statement is $\triangle ABC \sim \triangle DFE$). Instead, write the similarity statement first, then read the proportion directly from the vertex order.
- Assuming SSA is a valid similarity or congruence shortcut. It is not, in general — only AA, SAS (with the angle between the two given sides), and SSS are valid; if a problem gives two sides and a non-included angle, look for another angle relationship instead.
- Mislabeling opposite and adjacent sides in a trig ratio because the triangle is drawn in an unfamiliar orientation. Instead, always mark the reference angle first, then identify which leg touches it (adjacent) and which doesn't (opposite), regardless of how the triangle is rotated on the page.
- In the converse of the Pythagorean theorem, squaring the wrong side as the 'hypotenuse.' Instead, always square the two shortest sides and compare their sum to the square of the longest side, never the other way around.
- Forgetting that a negative scale factor in a dilation both resizes and reflects the figure through the center of dilation, not just resizes it — check the sign of k, not just its magnitude.
- Leaving a calculator in radian mode when computing inverse trig for degree-based problems, producing an answer that's numerically nonsense for a triangle angle. Always confirm degree mode before evaluating $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$.
Drill this unit until it sticks
These questions come back on a schedule built from what you get wrong, alongside the rest of Geometry. Free, and no account needed to start.