Olympiad Test: The Triangle and Its Properties - Free MCQ with solutions

• In a right triangle, the Pythagorean theorem states: the square of the hypotenuse equals the sum of the squares of the other two sides.
• Given: AB² = BC² + AC².
• This implies AB is the hypotenuse, as its square is the sum of the squares of BC and AC.
• In a right triangle, the right angle is opposite the hypotenuse.
• Thus, the right angle is at vertex C.

• The triangle is formed by sides BC = 7.2 cm and AC = 6 cm, with an angle ∠120° between them.
• An angle greater than 90° makes the triangle obtuse.
• The 120° angle is between sides BC and AC, meaning this angle is obtuse.
• Since one angle is greater than 90°, the triangle is classified as an obtuse-angled triangle.
• Hence, the correct answer is A: An obtuse angled triangle.

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

The Pythagorean theorem states that in a right-angled triangle:

• The side opposite the right angle is called the hypotenuse.
• The other two sides are known as the legs.

This property is essential for determining the relationship between the sides of a right-angled triangle.

If the sum of the squares of the lengths of the legs equals the square of the hypotenuse, then the triangle is confirmed as right-angled:

• Formula: a² = b² + c²

Thus, the triangle must be right-angled for the Pythagorean property to hold true.

Solution:

To find the value of x, we can use the angle sum property of a triangle, which states that the sum of the interior angles is always 180 degrees.

• Let the angles be represented as follows:
• x + x + x = 180 degrees
• This simplifies to:
• 3x = 180
• Now, divide both sides by 3:
• x = 180 / 3

Thus, the value of x is 60 degrees.

Solution:

The exterior angle of a triangle is equal to the sum of the two interior opposite angles. In this case:

• Given: Exterior angle = 120°
• Interior angles: 60° and x

We can express this relationship as:

120° = 60° + x

To find x, we rearrange the equation:

x = 120° - 60°

Thus, x = 60°.

Median is a line that connects a vertex of a triangle to the mid-point of the opposite side. This line divides the triangle into two equal areas.

In triangle PQR, the segment PD is the median because:

• It joins vertex A to the mid-point D of side BC.
• This segment ensures that the area on either side of the median is equal.

Thus, PD is confirmed as the median of triangle PQR.

Solution:

• Start with the equation: y + 125 = 180
• Solve for y:
  • y = 55
• Next, use the equation: x + y + 90 = 180
• Substituting y:
  • x + 55 + 90 = 180
• Simplifying gives:
  • x + 145 = 180
• Finally, solve for x:
  • 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:

(Leg)² + (Leg)² = (Hypotenuse)²

Since the legs are equal, we have:
x² + x² = 100
2x² = 100
x² = 100 / 2 = 50
x = √50 = √(25 × 2) = 5√2

Conclusion:
The length of each leg is 5√2 units.

Solution:

To find the value of x, we can use the angle sum property of a triangle, which states that the sum of the interior angles is always 180°.

• Given angles: 50° and 60°.
• Set up the equation: x + 50 + 60 = 180°.
• Simplify the equation: x + 110 = 180.
• Isolate x: x = 180 - 110.
• Calculate: x = 70°.

Thus, the value of x is 70°.

Solution:

In a triangle, if two sides are equal, the angles opposite those sides are also equal. This property applies to an isosceles triangle.

• Given that the two equal sides are opposite to angle x.
• Thus, angle x measures 40°.

Therefore, the measure of angle x is 40°.

Solution:

To find the length of AB in triangle ABC, which is right-angled at C, we use the Pythagorean Theorem.

• Given:
  • AC = 5 cm
  • BC = 12 cm
• According to the theorem:

  AB² = AC² + BC²

• Calculating:

  AB² = 5² + 12²

  AB² = 25 + 144

  AB² = 169

• Taking the square root:

  AB = 13 cm

Using the Pythagorean theorem, we can find the length of side QR in triangle PQR:

• Given:
• PQ = 3 cm
• PR = 4 cm
• Apply the theorem:
• QR² = PR² + PQ²
• QR² = 4² + 3²
• QR² = 16 + 9
• QR² = 25
• Calculate QR:
• QR = √25
• QR = 5 cm

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

In a right-angled triangle, the largest angle is always 90°.

Therefore, the side opposite the 90° angle is the longest side.

This side is known as the Hypotenuse.

The hypotenuse of a triangle is the longest side.

In triangle PQR, if angle P = 90°, then QR is the longest side.

Thus, QR is the longest side.

Solution:

In an isosceles triangle, where two sides are equal, the angles opposite these sides are also equal. This leads us to the following steps:

• Let the equal angles be 45° each.
• According to the angles sum property of triangles, the sum of the angles in a triangle is 180°.
• Thus, we can set up the equation: x + 45° + 45° = 180°.
• Simplifying this gives: x + 90° = 180°.
• Solving for x results in: x = 90°.

Therefore, the measure of angle x is 90°.

Solution:

In the given triangle, since two sides are equal, it is classified as an isosceles triangle with one angle measuring 90°.

Let the two equal angles be represented as x each.

According to the angle sum property of triangles:

• x + x + 90° = 180°
• 2x + 90° = 180°
• 2x = 90°
• x = 45°

Thus, the other two angles are each 45°.

A triangle in which all three sides are of equal lengths is called an equilateral triangle.

Key characteristics of an equilateral triangle include:

• All sides are of equal length.
• Each angle measures 60°.

When you take two copies of an equilateral triangle and place one on top of the other, they will fit perfectly regardless of how you rotate them. This demonstrates that equal side lengths result in equal angles.

Let the measures of the angles be 1x, 2x, and 7x.

We can calculate the angles as follows:

• Sum of angles in a triangle = 180°
• Equation: 1x + 2x + 7x = 180°
• Simplifying gives: 10x = 180°
• Thus, x = 18°

The angles of the triangle are:

• 1x = 18°
• 2x = 36°
• 7x = 126°

Since one angle is greater than 90°, 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.

Adding both sides, we have:
∠y + ∠1 + ∠2 = 180^ο (Angle Sum Property of a triangle)
∠x + ∠x + ∠x = 180^ο
3∠x = 180^ο
∠x = 180/3 = 60^ο

∠x = 60^ο, ∠y = 60^ο, ∠1 = 60^ο , ∠2 = 60^ο
Therefore, the value of all angles is 60° and this shows it is an equilateral triangle.

The sum of all angles in a triangle is always 180°.
40° + 60° + x = 180°
x = 180° - (40° + 60°)
x = 180° - 100° = 80°