Question

the sum of lengths of the perpendicular sides of a right triangle is 26 cm and its area is 84 cm².

a) What is the product of perpendicular sides ?

b) find lengths of the perpendicular sides

c) Find lenght of the hypotnuse​

Answer 1

Given :-

• the sum of perpendicular sides of a right triangle is 26 cm
• area of right triangle is 84 cm²

To find :-

1. the product of perpendicular sides ?
2. the length of the perpendicular sides ?
3. length of the hypotenuse ?

Formula used :-

$\sf \: area \: of \: right \: triangle \: = \frac{1}{2} \times base \times height \:$

$\sf \: hypotenuse \: of \: the \: right \: triangle \: \\ \sf \implies \: \sqrt{ {base}^{2} + {height}^{2} }$

Solution :-

• let base of the right triangle = x cm
• let height of the right triangle = y cm
• x + y = 26 ............. eq.(1)

ATQ

$\sf \: \dfrac{1}{2} \times x \times y = 84 \\ \\ \sf \: x \times y = 84 \times 2 \\ \\ \sf \: x \times y =168 \: cm......... \: eq.(2)$

• hence, the product of perpendicular sides is 168 cm

NOW,

from eq.(2)

$\sf \: x \times y = 168 \\ \\\sf \: x = \dfrac{168}{y} \\ \\ \sf \: put \: the \: value \: of \: \bf \: x = \dfrac{168}{y} \sf \: in \: eq.(1) \\ \\ \sf \: x + y = 26 \\ \\ \sf \dfrac{168}{y} + y = 26 \\ \\ \sf \dfrac{168 + {y}^{2} }{y} = 26 \\ \\ \sf \: 168 + {y}^{2} = 26y \\ \\ \sf \: {y}^{2} - 26y + 168 = 0 \\ \\ \sf \: {y}^{2} - (14 + 12)y + 168 = 0 \\ \\ \sf \: {y}^{2} - 14y - 12y + 168 = 0 \\ \\ \sf \:y(y - 14) - 12( - 14) = 0\\ \\ \sf \:(y - 14)(y - 12) = 0 \\ \\ \sf \: y - 14 = 0 \\ \\ \sf \:y = 14 \: cm \: \\ \\ \sf \: y - 12 = 0 \\ \\ \sf \:y = 12 \: cm \:$

when y = 14 cm

then x :-

x + y = 26

x = 26 - 14

x = 12 cm

when y = 12 cm

then x :-

x + y = 26

x = 26 - 12

x = 14 cm

therefore,

base of right triangle = 12cm or 14cm

also height of right triangle = 12cm or 14cm .

hypotenuse = ${\sf \:\sqrt{{base}^{2} + {height}^{2}}}$

hypotenuse = $\sf\: \sqrt{ {12}^{2} + {14}^{2} }$

hypotenuse = ${\sf\:\sqrt{144 +196}}$

hypotenuse = ${\sf\:\sqrt{340 }cm}$

here , I have taken base as 12 cm and height as 14 cm because triangle is right triangle . sides can't be equal .

we can also take base as 14 cm and height as 12 cm . but we can't take both side same .

Answer 2

Step-by-step explanation:

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