Question

part ii - constructed response
find the value of x in each right triangle. if necessary, round to the nearest tenth.
1.
right triangle with legs 3 and x, hypotenuse 5
2.
right triangle with legs 8 and 6, hypotenuse x

1. jovan is building a fort in his backyard using a 10 - foot board. he leans the base of the board 6 feet away from the base of the wall. how high will the board be able to reach up the wall? if necessary, round to the nearest tenth.
2. alpia biked 1 mile north from her home to the museum. she then biked 2 miles east from the museum to her friend’s house. how far will she bike in miles if she bikes home from her friend’s house in a straight line? if necessary, round to the nearest tenth.
3. a box measures 15 inches by 8 inches by 5 inches. what is the length of its longest diagonal? round to the nearest whole number.

Explanation:

Step1: Apply Pythagorean theorem

For right triangles, $a^2 + b^2 = c^2$, where $c$ is hypotenuse.

Problem 1:

Step1: Isolate $x$ term

$x^2 = 5^2 - 3^2$

Step2: Calculate and solve for $x$

$x^2 = 25 - 9 = 16$
$x = \sqrt{16} = 4$

Problem 2:

Step1: Substitute values into theorem

$x^2 = 6^2 + 8^2$

Step2: Calculate and solve for $x$

$x^2 = 36 + 64 = 100$
$x = \sqrt{100} = 10$

Problem 3:

Step1: Define variables, isolate height

Let $h$ = height, $h^2 = 10^2 - 6^2$

Step2: Calculate and solve for $h$

$h^2 = 100 - 36 = 64$
$h = \sqrt{64} = 8$

Problem 4:

Step1: Substitute into Pythagorean theorem

Let $d$ = straight distance, $d^2 = 1^2 + 2^2$

Step2: Calculate and solve for $d$

$d^2 = 1 + 4 = 5$
$d = \sqrt{5} \approx 2.2$

Problem 5:

Step1: 3D diagonal formula

For box, $d = \sqrt{l^2 + w^2 + h^2}$

Step2: Substitute and calculate

$d = \sqrt{15^2 + 8^2 + 5^2} = \sqrt{225 + 64 + 25} = \sqrt{314} \approx 18$

Answer:

1. $4$
2. $10$
3. $8$ feet
4. $2.2$ miles
5. $18$ inches