a) Find the area of triangle PQR correct to 3 decimal places. b) Hence find the length of the altitude from Q to [RP].
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Understand the Problem
The question asks for the area of triangle PQR, given the lengths of two sides and one angle. After computing the area, the question asks to find the length of the altitude from point Q to the side RP, using the area previously calculated.
Answer
Area of triangle PQR = $71.614 \, m^2$ Altitude from Q to RP = $8.425 \, m$
Answer for screen readers
Area of triangle PQR = $71.614 , m^2$ Altitude from Q to RP = $8.425 , m$
Steps to Solve
- Calculate the area of the triangle PQR
We are given two sides and the included angle, so we can use the formula: $$ \text{Area} = \frac{1}{2}ab\sin(C) $$ Where $a$ and $b$ are the lengths of two sides, and $C$ is the included angle
- Plug in the values
$a = 14$, $b = 17$, $C = 37^\circ$ $$ \text{Area} = \frac{1}{2}(14)(17)\sin(37^\circ) $$
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Calculate the sine and simplify $$ \text{Area} = \frac{1}{2}(14)(17)(0.6018) $$ $$ \text{Area} = 71.614 $$
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Find the altitude from $Q$ to $RP$ The area of a triangle can also be expressed as $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$ In this case, we consider $RP$ as the base and the altitude from $Q$ to $RP$ as the height
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Plug in the values to find the altitude (height)
Area = 71.614, base = 17 $$ 71.614 = \frac{1}{2} \times 17 \times \text{height} $$
- Solve for height $$ \text{height} = \frac{2 \times 71.614}{17} $$ $$ \text{height} = \frac{143.228}{17} $$ $$ \text{height} = 8.425 $$
Area of triangle PQR = $71.614 , m^2$ Altitude from Q to RP = $8.425 , m$
More Information
The area of a triangle can be found using the formula $Area = \frac{1}{2}ab\sin(C)$ when two sides and the included angle are known.
Tips
A common mistake is using the incorrect formula for the area of a triangle. Ensure you use the correct formula based on the given information. Also, rounding errors can occur if you round intermediate calculations; it's better to round only the final answer.
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