What this lesson covers
- 45-45-90 is 1 : 1 : root 2, and 30-60-90 is 1 : root 3 : 2, with root 3 always on the 60 degree side
- Divide the given side by the matching template side to get the scale factor, then multiply the whole template by it
- The hypotenuse of a 30-60-90 is twice the SHORT leg, never twice the long leg
- Equal angles force proportional sides, so a parallel cut inside a triangle is a similarity problem in disguise
Questions worked in the video
- 0:52A right triangle has a 30 degree angle and a hypotenuse of 12. Find both legs.
- 1:28A 45-45-90 triangle has a hypotenuse of 10. Find the leg.
- 1:55In triangle ABC, DE is parallel to BC. AD = 4, DB = 6, AE = 6. Find EC.
- 2:24A square has a diagonal of 10. Find the side length and the area.
Chapters
- 0:00Skip the Pythagorean theorem
- 0:25The two templates
- 0:52Scale, do not solve
- 1:28Half a square, and the triples
- 1:55Similar triangles
- 2:24Your turn
- 3:11Three traps
- 3:36Recap
Lesson transcript
The narration of the video, word for word, under its chapter headings.
0:00Skip the Pythagorean theorem
Welcome to SAT Climb. A right triangle, one thirty degree angle, hypotenuse twelve. Most students reach straight for the Pythagorean theorem. You do not have to. This is one of only two triangles the SAT reuses over and over, and both of its other sides are already decided.
0:25The two templates
There are only two special right triangles. Forty five, forty five, ninety: two equal legs, and a hypotenuse that is a leg times root two. Thirty, sixty, ninety: the short leg sits opposite the thirty, the long leg opposite the sixty is that short leg times root three, and the hypotenuse is exactly twice the short leg. Learn these two pictures and a whole family of questions collapses.
0:52Scale, do not solve
Here is the money move. Do not solve. Scale. Our triangle is a thirty, sixty, ninety, and the hypotenuse is twelve. In the template the hypotenuse is two, so the scale factor is twelve divided by two, which is six. Now multiply every side of the template by six. The short leg was one, so it is six. The long leg was root three, so it is six root three. Check it if you like. Six squared is thirty six, six root three squared is one hundred eight, and together they make one hundred forty four, which is twelve squared.
1:28Half a square, and the triples
Now the forty five, forty five, ninety, which is really just a square cut in half. If a leg is seven, the hypotenuse is seven root two. Run it backwards too. If the hypotenuse is ten, divide by root two and the leg is five root two. And keep the Pythagorean triples in the same drawer. Three four five. Five twelve thirteen. Eight fifteen seventeen. And every multiple of them, like six eight ten.
1:55Similar triangles
Special triangles are one case of a bigger idea. Similarity. Same angles means sides in proportion. Here, D E is parallel to B C, so the small triangle and the big triangle have identical angles. A D is four and D B is six, so the whole side A B is ten. A E is six. Set up four over ten equals six over A C. That gives A C equals fifteen, so E C is nine. Matching sides, one ratio.
2:24Your turn
Your turn, and this one hides a special triangle inside it. A square has a diagonal of length ten. Find the side length of the square, and then find its area. Pause the video here. The diagonal of a square cuts it into two forty five, forty five, ninety triangles, and that diagonal is the hypotenuse. Hypotenuse equals leg times root two, so ten equals s times root two, and s is ten over root two, which is five root two. The area is s squared. Five root two, squared, is twenty five times two. Fifty.
3:11Three traps
Three traps. One, putting root three on the short leg. Root three always goes with the sixty degree side, the longer one. Two, doubling the wrong side. The hypotenuse is twice the short leg, never twice the long leg. Three, matching sides before matching angles. In similar triangles, pair up the sides that sit opposite equal angles.
3:36Recap
Special right triangles, solved. Recognize the shape, find the scale factor, then multiply. Forty five, forty five, ninety hands you root two. Thirty, sixty, ninety hands you root three and a doubled hypotenuse. Start free at satclimb.com.
Read the written version: the Right triangles & trigonometry strategy guide, then try 25 hard Right triangles & trigonometry questions with full explanations. Both are free, no account needed.