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The sides of a triangle are 13 cm, 14 cm, and 15 cm. Find its area.
84 sq. cm
90 sq. cm
72 sq. cm
96 sq. cm
84 sq. cm
Use the property of Heron's triangles with consecutive sides (13, 14, 15). The height to the side of length 14 splits the base into segments of 5 and 9, making two right-angled triangles (5, 12, 13) and (9, 12, 15), thus height = 12 and Area = 0.5 * 14 * 12 = 84.
Sides of the triangle are a = 13 cm, b = 14 cm, and c = 15 cm.
Area=s(sтИТa)(sтИТb)(sтИТc)тАЛ┬аwhere┬аs=2a+b+cтАЛ
Use the property of Heron's triangles with consecutive sides (13, 14, 15). The height to the side of length 14 splits the base into segments of 5 and 9, making two right-angled triangles (5, 12, 13) and (9, 12, 15), thus height = 12 and Area = 0.5 * 14 * 12 = 84.
Many students miscalculate the semi-perimeter s or perform arithmetic errors under the square root sign, leading to incorrect area values.
Calculate the semi-perimeter
Calculate the semi-perimeter s by taking the average of the sides: s=(13+14+15)/2.
s=213+14+15тАЛ=21┬аcm
Apply Heron's Formula
Substitute the values of s,a,b, and c into Heron's formula: Area=21(21тИТ13)(21тИТ14)(21тИТ15)тАЛ.
Area=21├Ч8├Ч7├Ч6тАЛ
Simplify and solve
Simplify the product under the radical: Area=7056тАЛ, which evaluates to 84.
Area=7056тАЛ=84┬аcm2
A is correct because applying Heron's formula with sides 13, 14, and 15 yields an area of exactly 84 sq. cm.
This concept is fundamental for coordinate geometry problems involving finding the area of triangles formed by three points using the distance formula first to find side lengths.
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