The perimeter of a right triangle is 30 cm. if its hypotenuse is 13 cm then what are two sides (3)

For StudentsFor ParentsFor SchoolsSuccess StoriesOutcomesAI

Full Marks Mathematics 9>Heron's Formula>TEST YOUR SKILLS>Q 5

1. Area of Triangles:

(i) Area of a triangle when its base and height are known is calculated by using the formula: Area of triangle =12×Base×Height

(ii) If b denote the base and p the perpendicular of a right triangle, then the area of the triangle =12bp

(iii) For an isosceles right-angled triangle, each of whose equal side is a, we have the area =a22

(iv) For an equilateral triangle, each of whose side is a, we have the area =34a2

2. Heron's Formula:

If a, b, c denote the lengths of the sides of a triangle, then area =ss-as-bs-c, where s=a+b+c2

3. The area of a quadrilateral can be calculated by dividing the quadrilateral into triangles and using Heron's formula for calculating area of each triangle.

Did you know that the area of a quadrilateral can be calculated by dividing the quadrilateral into two triangles and using Heron's formula? Let's watch this...

Do you know how to calculate the area of a triangle when all three side lengths are given? It is very easy. Watch this video to understand how to calculate.

It is easy to find out the area of a triangle when we know its height and base. This video helps us understand the area of a triangle with examples. Let's wa...

We all love to go for a sail, but there are certain questions that we have never asked. Click and find out the application of Heron's formula in regards to a...

Can you determine the area of the sail of a boat? The conventional way to find sail area is to calculate the area of the fire triangle between the masthead s...

Uh-Oh! That’s all you get for now.

We would love to personalise your learning journey. Sign Up to explore more.

Skip for now

Uh-Oh! That’s all you get for now.

We would love to personalise your learning journey. Sign Up to explore more.

Skip for now

Answer

Verified

Hint: In this question first use the property that the perimeter of the triangle is the sum of all the sides, later on apply the concept of Pythagoras Theorem, so use these concepts to reach the solution of the question.Complete step-by-step answer:

Let the right angled triangle be ABC as shown in figure which is right angled at B.And it is given that the hypotenuse of the triangle is 13 cm.$ \Rightarrow AC = 13{\text{ cm}}$.Let the other two sides of the triangle be x and y respectively.$ \Rightarrow AB = x{\text{ cm, }}BC = y{\text{ cm}}$.Now, it is given that the perimeter of a right angled triangle is 30 cm.So as we know that the perimeter of a triangle is the sum of all sides.$ \Rightarrow AB + BC + CA = 30 \\ \Rightarrow x + y + 13 = 30 \\ \Rightarrow x + y = 30 - 13 = 17 \\ \Rightarrow x + y = 17{\text{ cm}}.............\left( 1 \right) \\ $Now as triangle ABC is a right angled triangle so, apply Pythagoras Theorem, we have${\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{Perpendicular}}} \right)^2} + {\left( {{\text{base}}} \right)^2}$${\left( {AC} \right)^2} = {\left( {AB} \right)^2} + {\left( {BC} \right)^2}$$ \Rightarrow {13^2} = {x^2} + {y^2} \\ \Rightarrow {x^2} + {y^2} = 169...........\left( 2 \right) \\ $Now as we know that ${\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy$Therefore from equation (1) and (2)$ \Rightarrow {17^2} = 169 + 2xy \\ \Rightarrow 2xy = 289 - 169 \\ \Rightarrow 2xy = 120 \\ \Rightarrow xy = 60..........\left( 3 \right) \\ $Now it is a known fact that ${\left( {x - y} \right)^2} = {\left( {x + y} \right)^2} - 4xy$Therefore from equation (1) and (3).$ \Rightarrow {\left( {x - y} \right)^2} = {\left( {17} \right)^2} - 4 \times 60 \\ \Rightarrow {\left( {x - y} \right)^2} = 289 - 240 = 49 \\ \Rightarrow x - y = \sqrt {49} = 7.........\left( 4 \right) \\ $Now add equation (1) and (4)$ \Rightarrow x + y + x - y = 17 + 7 \\ \Rightarrow 2x = 24 \\ $$ \Rightarrow x = 12$ cm.Now from equation (1)$ x + y = 17 \\ \Rightarrow 12 + y = 17 \\ \Rightarrow y = 17 - 12 = 5 \\ $$\therefore y = 5$ cm.So, the length of the other two sides are 12 and 5 cm.So, this is the required answer.Note: Whenever we face such types of questions always remember that the perimeter of any figure is sum of all the sides, so use this property and construct the equation, later on apply the property of Pythagoras theorem and construct another equation then solve these two equations as above we will get the required length of the other sides of the triangle.