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Question

A circle touches the side BC of $$ \Delta$$ ABC at P and touches AB and AC produced at Q and R respectively.
$$ If AQ = 5 cm$$, then find the perimeter of$$\Delta$$ ABC

83931_69e3ac807de44601b1a25b9a16123eb3.png


A
5cm
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B
10cm
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C
25cm
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D
50cm
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Solution

The correct option is D $$5 cm$$
$$AQ=5$$cm

Perimeter of $$\Delta ABC$$

$$AQ=AR=5$$cm

$$\{$$From a external point if drawn two tangents are equal in length$$\}$$
Perimeter of $$\Delta ABC=AB+BC+AC$$ .......$$(1)$$

But $$BQ=BP$$ .....$$(2)$$
$$CP=CR$$ ......$$(3)$$
$$\{$$From a external point if drawn tangents are equal in length$$\}$$

$$\Rightarrow BP+PC=BQ+CR=BC$$ ..........$$(4)$$

And $$AQ=AB+BQ=AB+BP$$ $$\{$$from equation $$(2)\}$$

$$\Rightarrow AB=AQ-BP=5-BP$$

$$\Rightarrow AB+BP=5$$

and $$AR=AC+CR=AC+CP$$  $$\{$$from equation $$(3)\}$$

$$AC=AR-CP=5-CP$$

$$AC=CP=5$$

From $$(1)$$

Perimeter$$=AB+BC+AC$$

$$=AB+BP+PC+AC$$ $$\{$$From equation $$(4)\}$$

$$=5+5$$

$$=10$$cm.

Mathematics

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