Olympiad Test: The Triangle and Its Properties - Class 7 MCQ

AB is the hypotenuse ⇒ C has the right angle.

Since ∠C=120^o>90^o, the triangle formed is obtuse.

In the Pythagoras property, the triangle must be a Right-angled triangle.

x + x+ x = 180 degrees [ angle sum property ]
3x = 180
x = 180/3 

x= 60 degree 

Exterior angle is equal to sum of interior opposite angles so,
120 = 60+x
x = 60^°

Median is a line joining the vertex and middle of the opposite side, and hence dividing the triangle in two equal parts. So PD is the median.

y + 125 = 180 
y = 55
and x+ y + 90 = 180
  x + 55 + 90 = 180 
x = 35

Let's solve the problem step by step:

Given:

• The two legs of a right-angled triangle are equal.
• The square of the hypotenuse is 100 square units.

Let the length of each leg be x units.

Using the Pythagorean theorem:

x + 50 + 60 = 180 (Angle sum property)
x + 110 = 180
x = 70

If in a triangle, 2 sides are equal then the angles opposite to them would also be equal
Hence , x = 40^ο

Acc to Pythagoras Theorem

AB²=AC²+BC²

AB²  = 5²+12²

=25+144 = 169
AB=13 cm

Using pythagoras theorem,
QR² = PR² + PQ²
4²+3²
=16+9=25
QR = 5cm

In a triangle, the side which is opposite to largest angle is the longest side.

Now, In a right - angled triangle, largest angle is always 90^∘.

So, the side which is opposite to 90^∘ is longest side.

Since, the side opposite to right - angle is known as "Hypotenuse".

Hypotenuse of a triangle is the longest side of a triangle.

In ΔPQR, if ∠P=90^∘ then QR is the longest side.

Hence, QR is the longest side.

In a triangle  △ABC, right-angled at B, the longest side is the hypotenuse, which is opposite the right angle.

Since the two sides are equal, the angles opposite to them are also equal.

So x + 45° + 45° = 180° (Angles sum property)

x = 90°

As two sides are equal so it is a isoceles triangle with one angle as 90^°

Let the two equal angles be x each 

Hence acc to Angle sum property

x + x + 90° = 180°

2x + 90° = 180°

2x = 90°

x = 45°

so other remaining two will be each 45^° 

A triangle in which all three sides are of equal lengths is called Equilateral Triangle.

 Let the measures of the angles be 1x,2x and 7x. We have, 1x+2x+7x=180^o⇒ x=18^o The angles are 18^o,36^o and 126^o.
∴ The triangle is obtuse angled.

According to the exterior angle property, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.