Triangles and the Pythagorean Theorem

Triangles carry more of this subtest than any other shape, because the Pythagorean theorem is the one piece of geometry that turns up inside other questions - in coordinate distance, in diagonals of rectangles, in Mechanical Comprehension.

Triangle types

By sides:

By angles: acute has all three under 90, right has exactly one 90, obtuse has one over 90. A triangle can have at most one angle of 90 or more, because the three must total 180.

The isosceles fact is the one that gets used. Equal sides face equal angles, so an isosceles triangle with a 40-degree apex has two base angles of (180 - 40) / 2, which is 70 each.

The Pythagorean theorem

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a squared plus b squared equals c squared, where c is the hypotenuse.

Two directions:

Finding the hypotenuse: add the squares of the legs, then take the root. Legs of 6 and 8 give 36 plus 64, which is 100, so the hypotenuse is 10.

Finding a leg: subtract. If the hypotenuse is 13 and one leg is 5, the other leg squared is 169 minus 25, which is 144, so the leg is 12.

Getting the direction wrong is the main error. The hypotenuse is the largest of the three, so if your answer for a leg comes out bigger than the hypotenuse, you added when you should have subtracted.

The triples

Memorize the first two and you will recognize most right triangles on the test without computing anything.

• 3-4-5 and its multiples: 6-8-10, 9-12-15, 12-16-20, 30-40-50.
• 5-12-13 and its multiples: 10-24-26, 15-36-39.
• 8-15-17.
• 7-24-25.

A triangle with legs 9 and 12 is a 3-4-5 scaled by 3, so the hypotenuse is 15. No squaring, no root.

The two special right triangles

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Two right triangles come up so often - 21 questions between them - that their side ratios are worth memorizing. Every side follows from the shortest one.

In a 45-45-90 triangle, one leg is 5. The hypotenuse is 5 root 2.

In a 30-60-90 triangle, the hypotenuse is 8. The shortest side, opposite the 30-degree angle, is half of that: 4.

Going the other way means dividing: a 45-45-90 triangle with a hypotenuse of 10 has legs of 10 / root 2, which simplifies to 5 root 2. In a 30-60-90 triangle, the side opposite 60 degrees is the short side times root 3, so if it is 9 root 3, the short side is 9.

An equilateral triangle cut in half is two 30-60-90 triangles. That is how its altitude (height) is found: half the side times root 3. A side of 2 gives an altitude of root 3.

Area

Area = one half x base x height.

The height is the perpendicular distance from the base to the opposite vertex.

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The slanted side is not the height. In a triangle with a base of 10, a slant side of 8 and a perpendicular height of 6, the area is one half times 10 times 6, which is 30. Using the 8 gives 40, and 40 is a choice.

In a right triangle the two legs are perpendicular to each other, so they are the base and the height. Area is one half times the product of the legs.

Perimeter is just the sum of the three sides, and needs no formula.

Similar triangles

Similar means same shape, different size: equal angles, and sides in proportion.

Triangles are similar if two angles match, which forces the third since all three sum to 180.

A triangle has sides 6, 8 and 10. A similar triangle has its shortest side 9. What is its longest side?

The scale factor is 9 divided by 6, which is 1.5. The longest side is 10 times 1.5, which is 15.

Or as a proportion: 6/10 equals 9/x, so 6x = 90 and x = 15.

Match the sides by position, not by order in the sentence. The shortest side of one corresponds to the shortest of the other. Pairing them wrongly is the error here, and the wrongly paired answer is a choice.

The classic application is a shadow problem:

A 6-foot pole casts a 4-foot shadow. A tower casts a 30-foot shadow at the same moment. How tall is the tower?

Same sun angle means similar triangles: 6/4 equals x/30, so 4x = 180 and x = 45 feet.

Keep heights on the same side of both fractions. Mixing them gives 20, which is a choice.

What you can skip

Across the 54 questions on this topic:

• Heron's formula never appears. Every area question gives a base and a height, or a right triangle.
• The law of sines and cosines never appear. Right triangles, the triples and the two special triangles carry every length question.
• Altitude-to-hypotenuse theorems. Two questions ask for the altitude drawn to a hypotenuse. If you meet one, the altitude equals the two legs multiplied, divided by the hypotenuse (6 x 8 / 10 = 4.8); do not memorize more.

Where people lose points

Adding when you should subtract in the Pythagorean theorem, or the reverse.

Applying the theorem to a triangle that is not right. It only works with a 90-degree angle.

Using a slant side as the height for area.

Forgetting the one half in the area formula. That gives exactly double, and double is always a choice.

Pairing similar sides wrongly.

Taking the root too early. a squared plus b squared gives c squared, and the root comes last.

Work one in under a minute

A rectangular field measures 60 by 80 yards. How long is the diagonal?

The diagonal splits the rectangle into two right triangles with legs 60 and 80.

Rather than squaring: 60 and 80 are 3-4-5 scaled by 20, so the diagonal is 5 times 20, which is 100 yards.

Check the long way: 3,600 plus 6,400 is 10,000, whose root is 100. Agreed, and the triple took three seconds.

Where this leads

The Pythagorean theorem is the engine behind the distance formula, the diagonal of any rectangle, and several Mechanical Comprehension questions about forces at an angle.