Triangles Questions

A triangle is a polygon that consists of three sides having three vertices. It is a two-dimensional figure. The interior angle of the triangle is measured to be 180 degrees. It is a polygon that has the least number of sides. 

Types of Triangles

Triangles are broadly classified based on the length of the sides and interior angles of the triangle.

Based on the sides, triangles are classified into three types, they are:

• Scalene Triangle.

• Isosceles Triangle.

• Equilateral Triangle.

• Scalene Triangle

A scalene triangle is a triangle where all three sides have different lengths. As the lengths of the sides are different, the interior angles are also different.  

• Isosceles Triangle

An isosceles triangle is a triangle where two out of three sides are equal. As the two sides are equal, the angles opposite to the equal sides are also equal. 

• Equilateral Triangle

An equilateral triangle is a triangle where all three lengths of the sides are equal. Such that, all the interior angles have an angle of 60 degrees since the sum of interior angles angle is 180 degrees. It is also referred to as an equiangular triangle.

Based on the interior angles, triangles are classified into three types, they are:

• Acute angle triangle

• Obtuse angle triangle 

• Right angle triangle

• Acute Angle Triangle

A triangle where all three interior angles are less than 90 degrees is called an acute-angled triangle. 

• Obtuse Angle Triangle

A triangle that has one of the three interior angles greater than 90 degrees is called an obtuse-angled triangle.

• Right Angle Triangle

A triangle where one of the three sides is exactly 90 degrees is called a right-angled triangle. The side opposite to this 90-degree angle has the longest side length, it is called the hypotenuse.

If any two interior angles of a triangle are similar to any two interior angles of another triangle, then the two triangles are said to be similar to each other.

For instance,

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Consider two triangles ΔABC and triangle ΔXYZ

If ∠ A = ∠X and ∠C = ∠Z 

then, ΔABC ~ ΔXYZ

Therefore,

AB/XY = BC/YZ = AC/XZ and ∠B = ∠Y

SAS or Side-Angle-Side Similarity

The SAS Similarity Theorem defines that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent then that two triangles will be termed as similar.

SSS or Side-Side-Side Similarity

If all the three sides of a triangle are equal to the three sides of another triangle, then the two triangles are similar.

For instance,

Consider two triangles ΔABC and triangle ΔXYZ

If AB/XY = BC/YZ = AC/XZ, 

then, ΔABC ~ ΔXYZ.

Therefore,

∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z

Solved Problems

1. If A ABC ~ ARPQ, AB = 3 cm, BC = 5 cm, AC = 6 cm, RP = 6 cm and PQ = 10 cm, then Find QR.

Solution:  ΔABC ~ ΔRPQ

So, AB/RP = BC/PQ = AC/RQ  as similar triangles have proportional sides

3/6=5/10=6/QR

½ = ½ 6/QR

QR= 12 cm

2. Find the Area of a Triangle Whose Two Sides are 12 cm and 16 cm and the Perimeter is 42cm.

Ans. Assume that the third side of the triangle is “x”.

Now, the three sides of the triangle are 12 cm, 16 cm, and “x” cm

It is given that the perimeter of the triangle = 36cm

So, x = 36 – (12 + 16) cm = 8 cm

∴ The semi perimeter of triangle = 36/2 = 18

Using Heron’s formula,

Area of the triangle,

\[= \sqrt{s(s-a)(s-b)(s-c)}\]

\[= \sqrt{18(18-12)(18-16)(18-8)}\] cm²

\[= \sqrt{18 \times 6 \times 2 \times 11}\]m²

\[= 18\sqrt{132}\] cm²

3. Corresponding Sides of Two Similar Triangles are in the Ratio of 1:2. If the Area of the Small Triangle is 16 sq. cm, then the Area of the Large Triangle is:

(a) 30 sq.cm.

(b) 6 sq.cm

(c) 7 sq.cm.

(d) 4 sq.cm

Answer: d

Solution: Let A1 and A2 are areas of the small and large triangle.

Then,

A2/A1=(side of large triangle/side of the small triangle)

A2/18=(1/2)2

A2=108 sq.cm.

4. Sides of Two Similar Triangles are in the Ratio 1: 9. Areas of these Triangles are in the Ratio

(a)2: 81

(b)1: 9

(c)1: 16

(d)16: 81

Answer: c

Explanation: Let ABC and DEF are two similar triangles, such that,

ΔABC ~ ΔDEF

And AB/DE = AC/DF = BC/EF = 1/9

As the ratio of the areas of these triangles will be equal to the square of the ratio of the corresponding sides,

∴ Area(ΔABC)/Area(ΔDEF) = AB2/DE2

∴ Area(ΔABC)/Area(ΔDEF) = (1/9)2 = 1/81 = 1: 81

Hence the above article is very useful as it provides solved examples that will help students to clear their doubts. Types of triangles are also explained in the article.