Question

please solve this ....urgent...​

Answer 1

Answer:

$\huge{\pink{\green{\sf{AREA_{3}=45.05cm^{2}}}}}$

Step-by-step explanation:

$\huge{\pink{\green{\underline{\red{\sf{SOLUTION-}}}}}}$

• In this question information is given about triangle and we have to find area of shaded region.

• According to this we have to first find Area of big triangle - Area of small triangle=Area of shaede region.

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{given} \\ \to \triangle \bold{ABC }\: is \: triangle \\ \to side \: of \: triangle \: are \bold{15cm \: each} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{to \: find} \\ \to area \: of \: shaded \: region = ?$

• According to given question :

$\to S= \frac{a + B+ C}{2} \\ \to S = \frac{17 + 15 + 15}{2} = \frac{47}{2} = 23.5\\ \to Area \: of \: triangle (ABC)= \sqrt{S(S - A)(S- B)(S - C)} \\ \to Area_{1} = \sqrt{23.5(23.5 - 17)(23.5 - 15)(23.5 - 15) } \\ \to Area_{1} = \sqrt{23.5 \times 6.5 \times 8.5 \times 8.5} \\ \to Area_{1} = \sqrt{11036.18} \\ \bold{\to Area_{1} = 105.05 {cm}^{2}} \\ \\ \to Side \: of \: small \: triangle(b) = ? \\ \therefore small \: triangle \: is \: right \: angled \: triangle \\ so,\: according \: to \: phythagoras \: theoram \\ \to {h}^{2} = {p}^{2} + {b}^{2} \\ \to {17}^{2} = {15}^{2} + {b}^{2} \\ \to 289 - 225 = {b}^{2} \\ \to {b}^{2} = 64 \\ \to b = \sqrt{64} \\ \to \bold{b = 8cm} \\ \\ \to Area \: of \: small \: triangle = \frac{1}{2} \times base \times height \\ \to Area_{2} = \frac{1}{2} \times 8 \times 15 \\ \to \bold{Area_{2} = 60 {cm}^{2} }$

• So we find both triangle areas.

$\to Area \: of \: shaded \: region= Area_{1} - area_{2} \\ \to Area_{3} = (105.05 - 60) {cm}^{2} \\ \bold {\to Area_{3} = 45.05 {cm}^{2}}$

Answer 2

Area of shades region = ar(ΔABC - ΔBDC)

Area of triangle ABC = 0.5 x b x h

(find h by Pythagoras)

$\sqrt{ {15}^{2} - ( {8.5})^{2} } \\ \sqrt{225 - 72.25 } \\ \sqrt{152.75 } \\ 12.36 \: cm$

Now,

Area = 0.5 x 17 x 12.36 = 105.06 cm^2

Area of BDC = 0. 5 x b x h

(find b by Pythagoras theorem)

You will get b = 8 cm

Now

Area = 0.5 x 15 x 8 = 60 cm^2

Area of shaded region = 105.06 - 60 =45.06 cm^2