Class 9 Maths

Triangles

Exercise 7.4

Question 1: Show that in a right angled triangle, the hypotenuse is the longest side.

Answer: In a right angled triangle, the angle opposite To the hypotenuse is 90°, while other two angles are Always less than 90°. As you know that the side opposite to the largest angle is always the largest in a triangle.

Question 2: In the given triangle sides AB and AC of Δ ABC are extended to points P and Q respectively. Also,
angle PBC < angle QCB. Show that AC > AB.

Triangle

Answer: `∠ABC=180°-∠PBC`
`∠ACB=180°-∠OCB`
Since `∠PBC <∠OCB`
So, `∠ABC >∠ACB`


As you know side opposite to the larger angle is larger than the side opposite to the smaller angle.
Hence, AC > AB

Question 3: In the given figure angle B < angle A and angle C < angle D. Show that AD < BC.

Triangle

Answer: `AO < BO` (Side opposite to smaller angle)
`DO < CO` (Side opposite to smaller angle)
So, `AO+DO < BO+CO`
Or, `AD < BC`

Question 4: AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD. Show that
angle A > angle C and angle B > angle D.

Quadrilateral

Answer: Let us draw two diagonals BD and AC as shown in the figure.
In ΔABD Sides AB < AD < BD

So, `∠ADB<∠ABD` --------(1)
(angle opposite to smaller side is smaller)
In ΔBCD sides `BC<DC<BD`
So, `∠BDC<∠CBD` ---------(2)

Adding equations (1) and (2)
`∠ADB+∠BDC<∠ABD+∠CBD`
Or, `∠ADC<∠ABC`

Similarly, in ΔABC
`∠BAC>∠ACD` ---------(3)
in ΔADC
`∠DAC>∠DCA` --------(4)

Adding equations (3) and (4)
`∠BAC+∠DAC&gat;∠ACB+∠DCA`
Or, `∠BAD>∠BCD`

Question 5: In following figure, PR > PQ and PS bisects angle QPR. Prove that angle PSR > angle PSQ.

Triangle

Answer: for convenience let us name these angles as follows:
`∠PQR=1 ∠PRQ=2 ∠QPR=3 ∠QPS=4`
`∠RPS=5 ∠PSQ=6 ∠PSR=7`
Since `PR>PQ`, so `∠1>∠2`
In ΔPQS
`∠1+∠4+∠6=180°`
In ΔPRS
`∠2+∠5+∠7=180°`
In both these triangles
`∠4=∠5`
`∠1>∠2`
So, for making the sum total equal to 180° the following will always be true:
`∠6<∠7`


Number System

Numbers are in the form of p/q or can be written in the form of p/q are called rational numbers.

Polynomials

Algebraic expression with many terms having variables and coefficients is called polynomial.

Coordinate Geometry

The distance of a point from the y - axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.

Linear Equations in Two Variables

A linear equation in two variables has infinite number of solutions.

Euclid's Geometry

The basic concept about a line is that it should be straight and that it should extend indefinitely in both the directions.

Lines & Angles

An angle is formed by two rays originating from the same end point. The rays making an angle are called the arms of the angle and the end-points are called the vertex of the angle.

Triangles

Two figures are congruent, if they are of the same shape and of the same size. Two circles of the same radii are congruent. Two squares of the same sides are congruent.

Quadrilaterals

Sum of the angles of a quadrilateral is 360°. A diagonal of a parallelogram divides it into two congruent triangles.

Parallelograms

In a parallelogram: opposite sides are equal, opposite angles are equal, and diagonals bisect each other

Circles

The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle.

Heron's Formula

Heron's formula is generally used for calculating area of scalene triangle.

Surface Area

How to calculate surface area of cylinder, cone and sphere?

Volume

How to calculate volume of cylinder, cone and sphere?

Statistics

Learn to make bar chart and histogram from given data.

Probability

Probability tells about likelihood of an event to occur.