ABC is drawn2 circumscribe a circle of radius 4cm such dat the segments BD and DC into which BC is divided D r of lengths 8cm and 6cm,Find AB nd AC.
You should come up with the notation for the following points: tangency point M lies on side AB and tangency point N lies on side AC.
Since the radius of inscribed circle is orthogonal to each side and it falls in tangency points D,M,N, hence, the segments `BM=BD=8 cm` and `CN=CD=6 cm` . Thus, the segments `AM=AN` .
`AM = AB-BM = AB-8`
`AN = AC-CN = AC-6`
You need to set equations above equal such that: `AB-8 = AC-6`
You need to use Pythagorean theorem in right triangles AMO and ANO such that:
`AO^2 = AM^2 + MO^2`
`AO^2 = AM^2 + 4^2 =gt AO^2 = AM^2 + 16`
`AO^2 = AN^2 + NO^2 =gt AO^2 = AN^2 + 16`
Notice that equating the relations above yields `AM=AN=a` .
You need to remember the relations between the radius of an incircle and the sides of triangle circumscribing the circle such that: ` r = sqrt(((p-AB)(p-BC)(p-AC))/p)`
`p = (AB+AC+BC)/2`
Substituting all the problem provides in relation above yields:
`4 = sqrt(((p-a-8)(p-14)(p-a-6))/p)`
`p = (a+8+a+6+14)/2 =gt p = (2a+28)/2 =gt p = a+14`
`4 = sqrt(((a+14-a-8)(a+14-14)(a+14-a-6))/(a+14))`
Reducing like terms yields:
`4 = sqrt(((6)(a)(8))/(a+14))`
You need to remove the square root, thus you need to raise to square both sides such that:
`16 = 48a/(a+14) =gt 16a + 16*14 = 48a`
`32a = 224=gt a = 224/32`
Hence, the side `AB = 8+7=15 cm ` and the side `AC = 6+7 = 13 cm.`
Hence, evaluating the lengths of sides yields `AB=15 cm ` and `AC = 13 cm.`
Let the given circle touch the sides AB and AC of the triangle at point E and F respectively and the length of the line segment AF be x.
CF = CD = 6cm (Tangents on the circle from point C)
BE = BD = 8cm (Tangents on the circle from point B)
AE = AF = x (Tangents on the circle from point A)
AB = AE + EB = x + 8
BC = BD + DC = 8 + 6 = 14
CA = CF + FA = 6 + x
2s = AB + BC + CA
= x + 8 + 14 + 6 + x
= 28 + 2x
s = 14 + x
Area of ΔOBC =
Area of ΔOCA =
Area of ΔOAB =
Area of ΔABC = Area of ΔOBC + Area ofΔOCA + Area of ΔOAB
Either x+14 = 0 or x − 7 =0
Therefore, x = −14and 7
However, x = −14 is not possible as the length of the sides will be negative.
Therefore, x = 7
Hence, AB = x + 8 = 7 + 8 = 15 cm
CA = 6 + x = 6 + 7 = 13 cm