Question

Could anyone please explain the shaded part of the solution? Why is it 1/2 ab?

Answer 1

Answer :

• It is 1/2ab because the area of a right angled triangle is given by , 1/2 x base x height and in the given triangle,the base is a and it's height is b making the area equal to 1/2ab.

Explanation :

Alright let's solve this my way

we are given that the sum of the three sides of a right angled triangle is 16cm and the sum of the square of the same sides is 98cm.

The sides here are termed as a,b, and c,

thus,

ATQ,

• a + b + c = 16.....(1)
• a^2 +b^2 + c^2 = 98....(2)

now, let's consider the fact that the square of the longest side or hypotenuse in a right angled triangle is equal to the sum of the square of the other two sides

thus,

• c^2 = a^2 + b^2 ....(3)

plugging in the value of c^2 in eq(2),

• a^2 + b^2 + a^2 + b^2 = 98
• 2(a^2 + b^2) = 98
• a^2 + b^2 = 98/2
• a^2 + b^2 = 49

thus,

• c^2 = a^2 + b^2 = 49 ....(4)

hence,

• c = √49
• c = 7.....(5)

plugging in the value of c in eq(1)

• a + b + 7 = 16
• a + b = 16 - 7
• a + b = 9 .....(6)

now,In order to find the area of the triangle,we will work out the value of ab (base x height of the triangle) using the identity:

• a^2 + b^2 = (a+b)^2 - 2ab
• 2ab = (a+b)^2 - (a^2 + b^2)

plugging in the value of (a + b) from eq (6) and that of (a^2 + b^2) from eq(4),

• 2ab = (9)^2 - 49
• 2ab = 32
• ab = 32/2
• ab = 16

hence,

• area = 1/2ab
• area = 1/2*16
• area = 8

therefore,the area of the triangle in cm^2 is 8.

Answer 2

Answer:

Area = 8 cm²

Step-by-step explanation:

The general formula for the area of a triangle is half the product of its base and height:

$\boxed{\begin{array}{l}\underline{\textsf{Area of a triangle}}\\\\A=\dfrac{1}{2}bh\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{ A is the area.}\\\phantom{ww}\bullet\;\textsf{ b is the base of the triangle.}\\\phantom{ww}\bullet\;\textsf{ h is the height of the triangle.}\end{array}}$

In the case of the given right triangle, its base is labelled a cm, and its height is labelled b cm. Therefore, if we substitute these values into the area formula, the area of this specific right triangle is given as:

$A=\dfrac{1}{2}ab$

We have been told that the sum of the three lengths of the triangle is 16 cm, and that the sum of the squares of the lengths of the three sides is 98 cm², so:

$a+b+c=16$

$a^2+b^2+c^2=98$

The Pythagorean Theorem states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of its legs. Since the legs of the given right triangle are labelled a and b, and the hypotenuse is labelled c, then:

$c^2=a^2+b^2$

Substitute this into the a² + b² + c² = 98 equation to eliminate the c² term:

$a^2+b^2+a^2+b^2=98$

Now, simplify:

$2(a^2+b^2)=98$

$a^2+b^2=49$

Since c² = a² + b², and we know that a² + b² = 49, then:

$c^2=49 \implies c=7$

Substitute c = 7 into the a + b + c = 16 equation to find the value of a + b:

$a + b + 7 = 16$

$a+b=9$

So, we now have two equations in terms of a and b:

$\begin{cases}a+b=9\\a^2+b^2=49\end{cases}$

Notice that a² + b² is the sum of two squares.

The general formula for the sum of two squares is:

$\boxed{\begin{array}{c}\underline{\textsf{The sum of two squares}}\\\\a^2+b^2=(a+b)^2-2ab\end{array}}$

This is derived from the algebraic identity (a + b)² = a² + 2ab + b².

We can rearrange this to isolate 2ab:

$2ab=(a+b)^2-a^2-b^2$

$2ab=(a+b)^2-(a^2+b^2)$

We determined that the equation for the area of the triangle is A = (1/2)ab. Rewrite 1/2 as the equivalent fraction 2/4:

$A=\dfrac{2}{4}ab$

$A=\dfrac{1}{4}(2ab)$

Now, we can substitute 2ab = (a + b)²- (a² + b²) so that we have an equation for the area of the triangle in terms of (a + b)² and a² + b²:

$A=\dfrac{1}{4}((a+b)^2-(a^2+b^2))$

To find the area of the triangle, all we have to do is substitute a + b = 9 and a² + b² = 49 into the new equation:

$A=\dfrac{1}{4}((9)^2-(49))$

$A=\dfrac{1}{4}(81-49)$

$A=\dfrac{1}{4}(32)$

$A=\dfrac{32}{4}$

$A=8$

Therefore, the area of the triangle is 8 cm².