Fill in the blanks with the correct numerical values based on the geometric properties des… — Armed Services Vocational Aptitude Battery (ASVAB)

Fill in the blanks with the correct numerical values based on the geometric properties described.

Answer:A city park features a right-angled triangular garden. The longest side of the garden measures

34\text{ yards}

, and one of the shorter sides is

16\text{ yards}

. The length of the other shorter side is 【30】 yards, and the area of the garden is 【240】 square yards.

The missing shorter side is 30 yards, and the area of the garden is 240 square yards.

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) with a hypotenuse of

34

and a leg of

16

, the missing leg is

\sqrt{34^2 - 16^2} = \sqrt{1156 - 256} = \sqrt{900} = 30\text{ yards}

. The area is found by multiplying half of the base by the height:

\frac{1}{2} \times 16 \times 30 = 240\text{ square yards}

Identify the given sides in relation to a right-angled triangle.

The hypotenuse (longest side)

c = 34\text{ yards}

, and one leg

a = 16\text{ yards}

. The other leg

b

is unknown.

Correctly identifying the hypotenuse is necessary to set up the Pythagorean theorem.

Use the Pythagorean theorem to find the length of the unknown leg.

a^2 + b^2 = c^2 \Rightarrow 16^2 + b^2 = 34^2 \Rightarrow 256 + b^2 = 1156 \Rightarrow b^2 = 900 \Rightarrow b = 30

The Pythagorean theorem relates the lengths of all three sides in a right triangle.

Calculate the area of the triangular garden.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 16 \times 30 = 240

The area of a right triangle is half the product of the lengths of its two perpendicular legs.

Using the Pythagorean theorem to find a missing leg, and applying the triangle area formula.

Question