Back to Exercise: Derive triangle area formula

Exercises: Derive the Formula A = (1/2)ab sin(C)

Grade 10·21 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-srt-d-9

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Recall / Warm-Up

A triangle has a base of 8 cm and a perpendicular height of 5 cm. What is its area?

40 cm²

20 cm²

13 cm²

80 cm²

In a right triangle, an acute angle is $\theta$ . The side opposite $\theta$ has length $h$ and the hypotenuse has length $a$ . Which equation is correct?

$\sin(\theta) = \dfrac{a}{h}$

$\sin(\theta) = \dfrac{h}{a}$

$\sin(\theta) = \dfrac{h}{a^2}$

$\cos(\theta) = \dfrac{h}{a}$

An altitude of a triangle is a line segment drawn from a vertex that is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

parallel to the opposite side

perpendicular to the opposite side (or its extension)

equal in length to the opposite side

bisecting the opposite side

Fluency Practice

In the derivation of $A = \dfrac{1}{2}ab\sin(C)$ , an altitude $h$ is drawn from vertex $B$ perpendicular to side $b$ (= $AC$ ). In the right triangle formed at vertex $C$ , the sine definition gives $\sin(C) = \dfrac{h}{a}$ . Solving for $h$ : $h =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . Substituting into $A = \dfrac{1}{2} \cdot b \cdot h$ gives $A =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

Find the area of a triangle with sides $a = 5$ cm and $b = 7$ cm and included angle $C = 60\degree$ . Use $\sin(60\degree) \approx 0.866$ . Round to the nearest tenth.

$Triangle PQR with sides PQ = 10, PR = 12, and included angle P = 50 degrees, with the area labeled as unknown$

Triangle $PQR$ has sides $PQ = 10$ and $PR = 12$ , with included angle $P = 50\degree$ . Find the area of triangle $PQR$ . Use $\sin(50\degree) \approx 0.766$ . Round to the nearest tenth.

A triangle has sides $a = 6$ m and $b = 8$ m with included angle $C = 120\degree$ . Find its area. Use $\sin(120\degree) \approx 0.866$ . Round to the nearest tenth.

Find the area of a triangle with two sides of length 4 ft and 9 ft and an included angle of 45°. Use $\sin(45\degree) \approx 0.707$ . Round to the nearest tenth.

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