The area of the triangle ABC is 320 cm²
Step-by-step explanation:
We can find the area of triangle ABC by using the sine rule of area
Area Δ ABC = [tex]\frac{1}{2}[/tex] (BC)(AC) sin(C)
The given is:
1. The length of BC is 34 cm
2. M is the mid-point of BC
3. MN ⊥ AC
4. AN = 25 cm , NC = 15 cm
∵ MN ⊥ AC
∴ ∠MNC , ∠ ANM are right angles
∵ M is the mid point of BC
∵ BC = 34 cm
∴ BM = MC = 34 ÷ 2 = 17
In ΔMNC
∵ m∠MNC = 90 ⇒ MN ⊥ AC
∵ MC = 17 cm ⇒ M is the mid-point of BC
∵ cos(C) = [tex]\frac{NC}{MC}[/tex] ⇒ cos(Ф) = adjacent/hypotenuse
∴ cos(C) = [tex]\frac{15}{17}[/tex]
- Find the measure of angle C by the inverse function [tex]cos^{-1}[/tex]
∴ m∠(C) = [tex]cos^{-1}\frac{15}{17}[/tex]
∴ m∠(C) = 28.07°
∵ Area Δ ABC = [tex]\frac{1}{2}[/tex] (BC)(AC) sin(C)
∵ AC = 25 + 15 = 40 cm
- Substitute the values of AC , BC and m∠C in the rule of the area
∴ Area Δ ABC = [tex]\frac{1}{2}[/tex] (34)(40) sin(28.07)
∴ Area Δ ABC = 320 cm²
The area of the triangle ABC is 320 cm²
Learn more:
You can lean more about the area of a triangle in brainly.com/question/4599754
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