Geometry: Similarity and Right-Triangle Trigonometry
Similar figures and the scale factor; dilations; AA, SSS and SAS similarity; proportions for a missing side and the side-splitter theorem; why area scales by k². Then the right triangle: opposite, adjacent and hypotenuse named from the angle; sine, cosine and tangent as ratios that depend only on the angle; solving for a side or an angle; angles of elevation and depression; the 45-45-90 and 30-60-90 triangles; measuring a tree with a tangent. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
From this corner: the side across the triangle is opposite, the side touching it is adjacent.
Opposite and adjacent are named FROM the angle you are using, and they trade places when you switch angles. The hypotenuse is fixed — always across from the right angle. Label the picture from the angle in the question before you pick sin, cos or tan.
What people get wrong
⚠️People often think…
Each side of a right triangle has one permanent name.
Only the hypotenuse does. Opposite and adjacent swap the moment you switch to the other acute angle, so a triangle labeled for one question is mislabeled for the next. Relabel from the angle you are actually using.
Relabel for every angle. It takes four seconds.
⚠️People often think…
cos 60° = 8/h means h = 8 × 0.5 = 4.
The unknown is on the BOTTOM, so you divide: h = 8 ÷ 0.5 = 16. And check it against the picture — a hypotenuse of 4 with a leg of 8 is impossible, because the hypotenuse is always the longest side.
The hypotenuse is always the longest side. Check it.
⚠️People often think…
A weird trig answer means you set the problem up wrong.
Check the calculator mode first. sin 30° is exactly 0.5; if your screen says −0.988, the calculator is in radians and every answer on the page is wrong for the same reason. Fix the mode, not the setup.
Test sin 30° before the test starts. It should read 0.5.
⚠️People often think…
A scale factor of 3 makes the area 3 times bigger.
Area scales by k², so 3 gives 9 times the area — because both the length AND the width tripled. Run it backwards too: areas of 5 and 45 mean k² = 9, so the scale factor is 3, not 9.
Lengths k, areas k², volumes k³.
Label it, then choose the ratio
1Watch one
cos 60° = 8/h. Find h.
cos 60° is 0.5.
So 0.5 = 8 ÷ h. The unknown is on the bottom.
Divide instead of multiply: h = 8 ÷ 0.5.
h = 16 — longer than the leg, as a hypotenuse must be.
2Do one with me
Two similar figures have a scale factor of 3. The small one has an area of 5.
Every length multiplies by
So the area multiplies by
The big figure's area is
💬One sentence, then you move on
Why does area scale by the square of the scale factor?
3Try one
Your calculator says sin 30° = −0.988. What is wrong? Answer in a word or two.
I want a hint first
sin 30° should be 0.5 every time. The problem is not the triangle — it is how the calculator is measuring the angle.
💬Last one — then you're done here
Why does the hypotenuse never change its name?
Where this goes
Where this lives
Finding the height of something you cannot climb — a tree, a roof, a flagpole — by measuring an angle and a distance you can reach.
What this feeds
Next unit is circles, area and volume — curved shapes, and the difference between covering one and filling it.
Name one height near you that you could measure without climbing it.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🧭 Solving a right-triangle problem — the routine
Draw the right triangle and mark the right angle. Mark the angle you know (or want). Label the side you know and the side you want RELATIVE TO THAT ANGLE: opposite (does not touch it), adjacent (touches it, not the hypotenuse), hypotenuse (across from the right angle). Pick the ratio that uses exactly those two sides — SOH, CAH or TOA. Write the equation, substitute, solve: multiply if the unknown is on top, divide if it is on the bottom, use the inverse if the unknown is the angle. Check: hypotenuse longest, angle under 45° means opposite shorter than adjacent, calculator in degrees.
Draw, mark, label, pick, solve, check.
🔑 The whole unit in one idea
Similar figures have the same shape, so every ratio of corresponding lengths is the same number — the scale factor. All right triangles with a given acute angle are similar (AA), so the ratios of their sides depend only on the angle. Those ratios have names: sine, cosine, tangent. Trigonometry is just similarity with a table of ratios attached, and every trig problem is a proportion in disguise.
Same angle, same shape, same ratio.
⚠️ Traps the test loves
Labeling “opposite” and “adjacent” from the wrong angle. They swap when you switch to the other acute angle; the hypotenuse never moves.
Multiplying when the unknown is in the denominator: cos 60° = 8/h gives h = 8 ÷ 0.5 = 16, not 8 × 0.5 = 4. A hypotenuse of 4 with a leg of 8 is impossible.
Calculator in radian mode. sin 30° = 0.5; if you see −0.99, switch to degrees.
Treating the angle of depression as the angle inside the triangle at the top. It is measured from the HORIZONTAL; the angle inside is 90° minus it — or use the equal angle of elevation at the bottom.
Scaling area by k instead of k². A scale factor of 3 gives 9 times the area; areas 5 and 45 mean a scale factor of √9 = 3.
Mixing positions in a proportion — 3/6 = x/5 instead of 3/6 = 5/x. Keep small over large on BOTH sides.
📐 The ratios and the values you should know — the chart
Ratio
Definition
30°
45°
60°
Other useful values
sine
opposite ÷ hypotenuse
0.5
≈ 0.71
≈ 0.87
sin 35° ≈ 0.57 · sin 40° ≈ 0.64 · sin 65° ≈ 0.91
cosine
adjacent ÷ hypotenuse
≈ 0.87
≈ 0.71
0.5
cos 35° ≈ 0.82 · cos 40° ≈ 0.77 · cos 65° ≈ 0.42
tangent
opposite ÷ adjacent
≈ 0.58
1
≈ 1.73
tan 20° ≈ 0.36 · tan 35° ≈ 0.70 · tan 40° ≈ 0.84
45-45-90
leg : leg : hypotenuse
—
x : x : x√2
—
legs 5 → hypotenuse 5√2 ≈ 7.07
30-60-90
short : long : hypotenuse
x
—
x√3
short leg 4 → long 4√3 ≈ 6.93, hypotenuse 8
🎯 How the test will ask
Two similar triangles with one side missing — write the proportion with matching positions, cross-multiply, check with the scale factor.
“Are these triangles similar? By which criterion?” — find the third angle first; AA needs only two matches.
A right triangle with one angle and one side — find another side. Label opposite/adjacent/hypotenuse from the given angle, then SOH-CAH-TOA.
Two sides, find the angle — build the ratio, then sin⁻¹, cos⁻¹ or tan⁻¹.
A word problem with a ladder, kite, ramp, shadow or angle of elevation — draw it first; the right angle is where vertical meets horizontal.
A 45-45-90 or 30-60-90 triangle with one side — use the ratios x : x : x√2 or x : x√3 : 2x; find the short leg first.
“How does the area change?” — by k², never by k.
✅ Before the test, can you…
Find a scale factor from two corresponding sides, and say which figure is the original?
Prove two triangles similar by AA, SSS~ or SAS~, and write the similarity statement in the right order?
Name opposite, adjacent and hypotenuse from either acute angle of a right triangle?
Write sin, cos and tan as ratios, and give their exact values at 30°, 45° and 60°?
Solve for a side whether the unknown is on top or on the bottom, and for an angle with an inverse function?
Draw and solve an angle-of-elevation problem, including the eye height?
Explain why sin 35° is the same in every right triangle with a 35° angle?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a surveyor: name the sides of a right triangle from the marked angle, decide which ratio each word problem needs, put a tree-height solution in order, then settle whether two triangles are similar. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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