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Heron's Formula: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

In the derivation, let $D=a^2+b^2-c^2$ . Which difference of squares equals $2ab-D$ ?

Answer choices for question 1 A $c^2+(a-b)^2$ B $(a+b)^2-c^2$ C $c^2-(a+b)^2$ D $c^2-(a-b)^2$
2

Three proposed lengths are positive, and $c$ is the largest. Suppose $s-c=0$ . What can you conclude about the proposed triangle?

Answer choices for question 2 A The shorter sides cannot reach each other even if stretched straight. B The two shorter sides together equal the longest, so the frame lies flat. C The triangle has positive area because all three lengths are positive. D The triangle is right because the product under the root is zero.
3

For a nondegenerate triangle with sides $a$ , $b$ , and $c$ , name two semiperimeter differences.

$y=s-b$

$z=s-c$

Which expression gives side $a$ ?

Answer choices for question 3 A $y+z$ B $y-z$ C $(y+z)/2$ D $s-y-z$
4

A nondegenerate triangle has a 24 cm side and a 39 cm side. Let $h_{24}$ and $h_{39}$ be the perpendicular heights to those sides. What is $\frac{h_{24}}{h_{39}}$ ?

Answer choices for question 4 A $\frac{8}{13}$ B $1$ C $\frac{13}{8}$ D $\frac{13}{16}$
5

A triangular panel has sides 25 cm, 26 cm, and 37 cm. A mount needs the perpendicular height to the 37 cm side to exceed 17 cm. Which conclusion follows?

Answer choices for question 5 A The height is exactly 17 cm. B The height exceeds 17 cm. C The height is less than 17 cm. D The three sides cannot determine whether the height exceeds 17 cm.
6

A triangle is to have perimeter 60 units. One side is 18 units and another is $b$ units. Which range gives exactly the values of $b$ that work for a nondegenerate triangle?

Answer choices for question 6 A $12\le b\le30$ B $0<b<42$ C $18<b<30$ D $12<b<30$
7

A new triangle is made by multiplying all three side lengths of a nondegenerate triangle by a positive number $t$ . What factor multiplies the original area in Heron's formula?

Answer choices for question 7 A $t$ B $t^4$ C $t^2$ D $\sqrt{t}$
8

A nondegenerate isosceles triangle has base 28 units, area 126 square units, and two equal sides of length $r$ . What exact length for $r$ follows from Heron's formula?

Answer choices for question 8 A $\sqrt{277}$ units B $\sqrt{115}$ units C $23$ units D $\sqrt{205}$ units
9

Each option lists proposed values of $s-a$ , $s-b$ , and $s-c$ , in that order. A nondegenerate triangle has semiperimeter 30 units.

Which option can occur?

Answer choices for question 9 A 0, 14, and 16 units B 9, 10, and 12 units C 8, 9, and 13 units D −1, 15, and 16 units
10

For a nondegenerate triangle, write $D=a^2+b^2-c^2$ . Why are both factors $2ab-D$ and $2ab+D$ positive?

Answer choices for question 10 A
The first factor equals $4(s-a)(s-b)$ .

The second equals $4s(s-c)$ .

All four semiperimeter factors are positive.
B Their product is positive, and two numbers with positive product must each be positive. C The difference $D$ is positive for every triangle. D The product $2ab$ is positive, so adding or subtracting any $D$ leaves a positive number.
11

Two valid triangular patches have these side lengths in meters.

Patch A has sides 16, 22, and 24.

Patch B has sides 16, 22, and 30.

Both areas round to 171 square meters. How do the exact areas compare?

Answer choices for question 11 A Patch A has the greater exact area. B The exact areas are equal. C The side lengths cannot settle the exact comparison. D Patch B has the greater exact area.
12

Three proposed lengths $a$ , $b$ , and $c$ are positive, with $a<b$ . Set $D=a^2+b^2-c^2$ . If $2ab-D=0$ , what follows?

Answer choices for question 12 A The lengths give a flat arrangement with $c=a+b$ . B The lengths give a flat arrangement with $b=a+c$ . C The lengths form a nondegenerate right triangle. D The lengths leave a gap because $b>a+c$ .

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