measure less than 50°. But this would make the total less than 180°. Therefore, x must be greater than 50, and Statement I is true. Eliminate the second choice. Statement II: The shortest side of a triangle will always be opposite the angle of smallest measure. Two of the angles have measure x; the third, ∠CBA, has a measure less than that: x – 15. Since
∠CBA is the smallest angle, the side opposite it, side AC, must be the shortest side. Since CB has length 10, AC must be less than 10. Statement II is true. Eliminate the first and fourth choices. Statement III: ∠ACB and ∠BAC both have a measure of x°, so ∆ABC is isosceles. Therefore, AB has the same length as CB (they’re opposite the equal angles). Since BC has a length of 10, AB must also have a length of 10. Statement III is not true. Statements I and II only must be true.
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the length of AC using the Pythagorean theorem.
The area of any right triangle equals one–half the product of the legs. If BC has a length of x, then AB has a length of 3x. (If BC is one–third the length of AB, then AB is three times the length of BC.) The area of the triangle is one-half their product, or (x)(3x).
This equals 6.
BC has a length of 2. So AB, which is 3x, is 6. Now use the Pythagorean theorem to find
AC.
Draw yourself a diagram so that the picture is more clear: In an isosceles right triangle, both legs have the same length. So,
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We’re given that the area is 32, so we can set up an equation to solve for : Remember, the ratio of the length of the legs to the length of the hypotenuse in any
isosceles right triangle is 1:
Since the legs have a length of 8, the hypotenuse is times 8, or 8 . An alternative is to use the Pythagorean theorem to find the hypotenuse:
4
If ∠DBA has a measure of 60°, ∠CBD, which is supplementary to it, must have a measure of 180 – 60, or 120°. ∠DCB has a measure of 30°; that leaves 180–(120 + 30), or 30
degrees for the remaining interior angle: BDC. Since ∠BCD has the same measure as ∠BDC, ∆BCD is an isosceles triangle, and the sides opposite the equal angles will have equal lengths. Therefore, BD must have the same length as BC, 4.
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2
(Note that we’ve added point D for clarity.) The area of a triangle is × base × height. If we treat AB as the base of ∆ABC, then the triangle’s height is CD. Each square has side 1, so we can just count the squares. AB = 1, CD = 4, so the area is × 1 × 4 = 2
We know one of the sides we’re given is the hypotenuse; since the hypotenuse is the longest side, it follows that it must be the larger value we’re given. The side of length must be the hypotenuse, since d is positive (all lengths are positive), and of a positive
value is always greater than of a positive value. Now we can use the Pythagorean theorem to solve for the unknown side, which we’ll call x.
Another way we can solve this, that avoids the tricky, complicated algebra is by picking a number for d. Let’s pick a number divisible by both 3 and 4 to get rid of the fractions: 12 seems like a logical choice. Then the two sides have length , or 4, and , or 3.
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If one of these is the hypotenuse, that must be 4, and 3 must be a leg. Now use the
Pythagorean Theorem to find the other leg: Now plug in 12 for d into each answer choice; the one which equals is correct.
30
Drawing a diagram makes visualizing the situation much easier. Picture a ladder leaning
against a building:
This forms a right triangle, since the side of the building is perpendicular to the ground. The length of the ladder, then, is the hypotenuse of the triangle; the distance from the foot
of the building to the base of the ladder is one leg; the distance from the foot of the building to where the top of the ladder touches the wall is the other leg. We can write
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these dimensions into our diagram:
The one dimension we’re missing (what we’re asked to find) is the length of the ladder, or the hypotenuse. Well, we could use the Pythagorean Theorem to find that, but these numbers are fairly large, and calculating will be troublesome. When you see numbers this large in a right triangle, you should be a little suspicious; perhaps the sides are a multiple of a more familiar Pythagorean Triplet. One leg is 18 and another leg is 24; 18 is just 6 × 3, and 24 is just 6 × 4. So we have a multiple of the familiar 3-4-5 right triangle. That means that our hypotenuse, the length of the ladder, is 6 × 5, or 30.
6
This problem involves as much algebra as geometry. The Pythagorean theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse, or, in this case:
and from here on in it’s a matter of algebra:
When the product of two factors is 0, one of them must equal 0. So we find that
According to the equation, the value of x could be either 0 or 6, but according to the diagram, x is the length of one side of a triangle, which must be a positive number. This means that x must equal 6 (which makes this a 6:8:10 triangle). Another way to do this problem is to try plugging each answer choice into the expression for x, and see which one gives side lengths which work in the Pythagorean Theorem. Choice (1) gives us 6, 8, and 10 (a Pythagorean Triplet) for the three sides of the triangle, so it must be the answer.