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

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

Solution

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