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Zorluk: OrtaPerimeter and Area of Plane Shapes

A right-angled triangle has a perimeter of 60 cm60\text{ cm} and a hypotenuse of length 25 cm25\text{ cm}. What is the area of the triangle in cm2\text{cm}^2?

Cevap: 150 cm²

Cevap

150
Let the perpendicular sides of the right-angled triangle be aa and bb, and the hypotenuse be c=25 cmc = 25\text{ cm}. From the perimeter, a+b+25=60a + b + 25 = 60, so a+b=35 cma + b = 35\text{ cm}. By the Pythagorean theorem, a2+b2=252=625a^2 + b^2 = 25^2 = 625. Squaring both sides of a+b=35a + b = 35 yields (a+b)2=a2+b2+2ab=352=1225(a + b)^2 = a^2 + b^2 + 2ab = 35^2 = 1225. Substituting a2+b2=625a^2 + b^2 = 625 gives 625+2ab=1225  ⟹  2ab=600  ⟹  ab=300625 + 2ab = 1225 \implies 2ab = 600 \implies ab = 300. The area of the right-angled triangle is 12ab=12×300=150 cm2\frac{1}{2}ab = \frac{1}{2} \times 300 = 150\text{ cm}^2.

Adım Adım Çözüm

1
Determine the sum of the two legs from the perimeter
a+b=35 cma + b = 35\text{ cm}
The perimeter of the triangle is a+b+c=60 cma + b + c = 60\text{ cm}, where the hypotenuse c=25 cmc = 25\text{ cm}.
2
Use the Pythagorean theorem for the sum of squares of the legs
a2+b2=625a^2 + b^2 = 625
In any right-angled triangle with hypotenuse 25 cm25\text{ cm}, a2+b2=252=625a^2 + b^2 = 25^2 = 625.
3
Expand (a+b)2(a + b)^2 to find the product of the legs abab
1225=625+2ab  ⟹  2ab=600  ⟹  ab=3001225 = 625 + 2ab \implies 2ab = 600 \implies ab = 300
Using the algebraic identity (a+b)2=a2+b2+2ab(a + b)^2 = a^2 + b^2 + 2ab allows determining abab directly without solving for individual side lengths.
4
Calculate the area of the right-angled triangle
\text{Area} = 150\text{ cm}^2
The area of a right-angled triangle with perpendicular sides aa and bb is given by 12ab\frac{1}{2}ab.

Anahtar Kavram

Perimeter and Area of Right-Angled Triangles using Algebraic Identities
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Konular

Perimeter and Area of Plane Shapes

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