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Concept:

The sum of all angles of a triangle is 180° = π 

Calculations:

Given, one of the angles of a triangle is 1/2 radian and the other is 99°.

⇒ \(\rm ∠ A = (\dfrac 1 2)^c\) and ∠B = 99° 

Convert the ∠B  into radians by multiplying by \(\rm \dfrac {π}{180^\circ}\).

⇒ \(∠ B = 99^\circ \times \dfrac {π}{180^\circ}\) = \(\dfrac {11π}{20}\)

As we know, the sum of all angles of a triangle is 180° = π

⇒ \(\rm ∠ A + ∠ B + ∠ C = π\)

⇒ \(\rm ∠ C = π - (∠ A + ∠ B)\)

⇒ \(\rm ∠ C = π - (\dfrac 12 + \dfrac{11π}{20})\)

⇒  \(\rm ∠ C = π - (\dfrac {10 + 11π}{20})\)

⇒  \(\rm ∠ C = \dfrac {9π -10}{20}\)

Hence, one of the angles of a triangle is 1/2 radian and the other is 99°, then the third angle in radian measure is \(\rm ∠ C = \dfrac {9π -10}{20}\)

Concept:         

The sides of an obtuse triangle should satisfy the condition that the sum of the square of two smaller sides should be less than the square of the largest side.

If a, b, c are in AP than \(b = \frac{{a + c}}{2}\)

If a, b, c are in GP than \(b = \sqrt {ac} \)

We know that,

Sum of all the angles in a triangle =180°

Let us consider the acute angle as x

Hence,

x + 90° + 70° = 180°

x + 160° = 180°

x = 180° – 160°

We get,

x = 20°

Therefore, the acute angle is 20°

Concept:

The centroid of a Triangle: The point where the three medians of the triangle intersect.

Let’s consider a triangle. If the three vertices of the triangle are A(x1, y1), B(x2, y2), C(x3, y3)

The centroid of a triangle

=\(\rm (\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})\)

Calculation:

Here, vertices of triangle (4,x), (y,-5) and (7, 8) and Centroid = (3, 5)

So, 3 = \(\rm \frac{4+y+7}{3}\)

⇒11+y = 9 

⇒ y = -2

And, 5 = \(\rm \frac{x-5+8}{3}\)

⇒ x + 3 = 15

⇒ x = 12

∴ x = 12, and y = -2

Hence, option (3) is correct. 

Given:

Perpendicular of right angled triangle = 8 cm

Area = 20 cm²

Formula used:

Area of right angled triangle = (1/2) × perpendicular × base

Calculation:

⇒ 20 cm² = (1/2) × 8 × base

⇒ base = 20/4

⇒ 5 cm

∴ The length of base is 5 cm

Given that BD ⊥ AC; CE ⊥ AC 

BD = CE 

Now in ΔABD and ΔACE 

∠ADB = ∠AEC (∵ given 90°) 

∠A = ∠A (commori angle)

BD = CE 

∴ ΔABD = ΔACE (∵ AAS congruence)

⇒ AB = AC (∵ C.P.C.T)

Triangle: A triangle is a loop in which three sides are joined together. The point at which lines are joined is called the vertex. The angle measure of a triangle is 180 degrees. The minimum criteria to construct a triangle is to have a length of two sides and an angle between them.

Properties of Triangle:

• The angle sum of a triangle is 180 degrees.
• An acute angle is an angle whose measure is less than 90°
• An obtuse angle is an angle whose measure is greater than 90°
• So, ∠A + ∠B + ∠C = 180°

Hence, the minimum criteria to construct a triangle is to have a length of two sides and an angle between them.

It is given that ∠EPA = ∠DPB

∠EPA + ∠DPE = ∠DPB + ∠DPE

∠DPA = ∠EPB

In ΔDAP and ΔEBP,

∠DAP = ∠EBP (Given)

AP = BP (P is mid-point of AB)

∠DPA = ∠EPB (From above)

∴ ΔDAP ≅ ΔEBP (ASA congruence rule)

AD = BE (By CPCT)