Geometry: Triangles
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1. What is the perimeter of an isosceles right triangle with an area of 18?

(A) 6
(B) 12
(C) 18
(D) 12√2
(E) 6 · (2 + √2)

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Since area is one-half base times height, the base times the height must be 2·18=36. The triangle is isoceles so the base and the height are equal so they are both √36=6. It's a right triangle so the two legs are each 6 and the hypotenuse is 6√2. The perimeter is 6+6+6√2 or (E).

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2. In triangle ABC, AB is chosen as the base. The height of the triangle with AB as the base is h and h = AB. Which of the following is true?

I. ABC could be a right triangle.
II. Angle C cannot be a right angle.
III. Angle C could be less than 45°.

(A) none
(B) I, only
(C) I and II, only
(D) I and III, only
(E) I, II, and III

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A right triangle with two equal legs satisfies I, so it is true. If AB is the base, the height is drawn from point C to AB. If angle C is 90 degrees then AB is the hypotenuse which is always longer than the height so II is true, there's no way C could be a right angle and still have h=AB. Draw base AB from left to right. Then draw a line parallel to AB a distance h=AB from AB. If you place point C on the parallel line, you'll have a triangle that fits the problem. If point C is far to the right or left, then angle C will be very small. So III is true and the answer is (E). Note that the height of a triangle need not be inside the triangle.

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3. If triangle XYZ is not a right triangle, then a line drawn from vertex Z to side XY that is perpendicular to XY is always:

(A) longer than XY
(B) shorter than XY
(C) longer than XZ and longer than YZ
(D) shorter than XZ and shorter than YZ
(E) none of these

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The shortest distance between a point and a line is a segment through the point perpendicular to the line. Since it's not a right triangle you know that XZ and YZ are not perpendicular to XY. So the line from Z to XY has to be shorter than both XZ and YZ. The answer is (D).

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4. The length of each of the sides of a triangle is a positive integer and n is a positive integer. If the length of one side is n + 10 and the length of another side is n + 12, then the shortest possible length of the third side is:

(A) n–2
(B) n+2
(C) n+11
(D) 2
(E) 3

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Suppose n is 5. Then we've got a 15 and a 17 so you know the third side has to be greater than 2 (in a triangle, the sum of any two sides has to be more than the other side). Since the length of each side is a positive integer, the shortest possible length of the third side is 3. No matter what you pick for n, you'll always get 3. The answer is (E).

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5. A triangle is inscribed in a circle whose radius is 2 centimeters. Each vertex of the triangle is on the circle and one side of the triangle is a diameter of the circle. What is the largest possible area of the triangle in square centimeters?

(A) 2
(B) 4
(C) 6
(D) 8
(E) 12

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If you draw this, you can see that the base is set at 4 (the diameter) and the largest possible height is 2 (the radius). So the largest possible area is one-half base times height or 4. The answer is (B).

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6. Triangle ABC is equilateral and BC=1. Another equilateral triangle is to be inscribed in ABC so that each side of triangle ABC contains one vertex of the inscribed triangle. Which of the three points along BC in the figure may be used as a vertex of the inscribed triangle?

(A) y, only
(B) y and z, only
(C) x and z, only
(D) x and y, only
(E) x, y, and z

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SATan loves equilateral triangles. SATan also loves symmetry. If you draw from point x to a point 3/4 of the way from C to A and then draw from than point to a point 3/4 of the way from A to B, you'll get a triangle where all the sides have to be the same (if there were a 'long' side, which would it be?). This argument works for all three points so the answer is (E).

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7. Triangle ABC is equilateral and BC = 1. A line segment is to be drawn from point D to AC. If the line segment intersects AC at point E, what is the shortest possible length of DE?

(A) 1 / 8
(B) 1 / 4
(C) 1 / 2
(D) √3 / 4
(E) √3 / 8

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If DE is the shortest possible then it must be perpendicular to AC. Since angle C is 60 (ABC is equilateral), triangle DCE is 30-60-90 wih hypotenuse DC. The short leg (CE) is half the hypotenuse or 1/8 and the long leg (DE) is √3 times the short leg or √3/8. The answer is (E).

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Note: Figure not drawn to scale.

8. Triangle ABC is equilateral. If BE = DE and DE is perpendicular to BC, what is the ratio of the length of CE to the length of BC?

(A) 1 / 3
(B) 1 / 4
(C) 1 / √3
(D) 1 / (√3 + 1)
(E) 1 / (√3 + √2)

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Triangle DCE is 30-60-90 and triangle DBE is 45-45-90. If CE is x units long then DE is x√3 (30-60-90 rules). This means BE is also x√3 (45-45-90 rules). So BC is x + x√3. Now you can do the ratio (the x will cancel). You get CE/BC = x/(x+x√3) = 1/(1+√3) or (D).

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Note: Figure not drawn to scale.

9. Triangle ABC is equilateral. If triangle DEF is also equilateral and DE is perpendicular to BC, what is the ratio of the length of CE to the length of BC?

(A) 1 / 3
(B) 1 / 4
(C) 1 / √3
(D) 1 / (√3 + 1)
(E) 1 / (√3 + √2)

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Triangles DCE and EBF are both 30-60-90. If CE is x then DE is x√3. This means FE is also x√3 which means FB must be x (you could say this is true because of symmetry or because the three 30-60-90 triangles have to be congruent). No matter how you slice is CE = FB = x. This means BE is 2x (it's the hypotenuse of a 30-60-90). So BC is 3x and the ratio of CE to BC is one-third. The answer is (A).

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Note: Figure not drawn to scale.

10. Triangle ABC is still equilateral. If M is the midpoint of BC and MP is perpendicular to AC and MQ is perpendicular to AB, what is the ratio of the length of PQ to the length of BC?

(A) 1 / 3
(B) 3 / 8
(C) 2 / 3
(D) 3 / 4
(E) cannot be determined

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There are a lot of ways to do this one. The easiest is to first realize that if BC = y, then MB = y/2 and BQ = y/4 (the short leg is half the hypotenuse in a 30-60-90 triangle). The next thing to realize is that PQ is parallel to BC so angles APQ and AQP are both 60 so triangle APQ is equilateral. Let's say PQ = x. This means AQ is also x. We're looking for x/y (that's the ratio the problem asks for). We have a nice equation: y = x + y/4. Dividing everything by y and subtracting 1/4 gets you x/y = 3/4. So the answer is (D).

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