Question

circle o is inscribed in the given triangle. what is the perimeter of the triangle?
options: 22 units, 30 units, 44 units, 60 units
(image shows a triangle with incircle, tangency points labeled with segments 4, 6, 12)

Explanation:

Step1: Recall the property of tangents

Tangents from a common external point to a circle are equal in length. Let the triangle be \( \triangle ABC \) with incircle touching the sides at \( P, Q, R \). Let the lengths of the tangents be: from one vertex (say \( A \)) to the points of tangency \( P \) and \( R \) be \( x \), from vertex \( B \) to \( P \) and \( Q \) be \( y \), and from vertex \( C \) to \( Q \) and \( R \) be \( z \). From the diagram, we see that one tangent segment is 4, another is 6, and another is 12. Wait, actually, looking at the diagram, let's label the triangle's sides. Let the side with length 12 have a tangent segment of length 4 (from the left vertex to \( P \)) and the side with length 6 (from the bottom vertex to \( Q \)). So, using the tangent - segment property: if we let the tangent from the top vertex to \( P \) be \( a \), from the top vertex to \( Q \) be \( b \), from the left vertex to \( P \) be 4, from the left vertex to \( R \) be 4 (since tangents from left vertex are equal), from the bottom vertex to \( Q \) be 6, from the bottom vertex to \( R \) be 6 (since tangents from bottom vertex are equal). And we know that the side with length 12 is composed of \( a + 4\), and the side with length (let's say the other non - 6, non - 12 side) is composed of \( b+6\), and the side with length 6 is composed of \( b \) (wait, no, let's re - express).

Wait, more clearly: Let the triangle have three sides. Let the lengths of the tangents from each vertex to the points of tangency be: Let the left vertex (with tangent length 4) have tangents of length 4 to both \( P \) and \( R \). The bottom vertex (with tangent length 6) has tangents of length 6 to both \( Q \) and \( R \). The top vertex has tangents of length \( x \) to both \( P \) and \( Q \). We know that one of the sides (the left - top side) is \( x + 4=12\), so we can solve for \( x \): \( x=12 - 4 = 8\).

Step2: Calculate the lengths of all sides

- The length of the side with tangent \( x \) and 6: \( x+6=8 + 6=14\)? Wait, no, wait. Wait, the three sides of the triangle are:
- Side 1: \( 4 + 12=16\)? No, no. Wait, let's use the standard formula for the perimeter of a triangle with an incircle. If we have a triangle with an incircle, and the lengths of the tangent segments from the vertices are \( a\), \( b\), \( c\), then the sides of the triangle are \( a + b\), \( b + c\), \( c + a\), and the perimeter \( P=(a + b)+(b + c)+(c + a)=2(a + b + c)\).

From the diagram, we can see that two of the tangent segments are 4 and 6, and one of the sides (the one with length 12) is composed of 4 and another tangent segment (let's call it \( m\)). So \( 4 + m=12\), so \( m = 8\). Now, the three tangent segments from the three vertices are 4, 6, and 8.

Then the sides of the triangle are:
- Side 1: \( 4+6 = 10\)? No, wait, no. Wait, the correct way is: If we have a triangle with an incircle, and the tangents from vertex \( A\) are length \( x\), from vertex \( B\) are length \( y\), from vertex \( C\) are length \( z\), then the sides are \( x + y\), \( y+z\), \( z + x\). The perimeter is \( 2(x + y+z)\).

From the diagram, we can see that one tangent segment is 4 (let's say from vertex \( A\) to the point of tangency on side \( AB\) and \( AC\)), one is 6 (from vertex \( B\) to the point of tangency on side \( AB\) and \( BC\)), and the third tangent segment (from vertex \( C\) to the point of tangency on side \( AC\) and \( BC\)) can be found from the side that has length 12. The side with length 12 is composed of the tangent from \( A\) (length 4) and the tangent from \( C\) (length \( t\)), so \( 4 + t=12\), so \( t = 8\).

Now, the three tangent lengths are \( x = 4\), \( y = 6\), \( z = 8\).

The sides of the triangle are:
- \( x + y=4 + 6 = 10\)
- \( y + z=6 + 8 = 14\)
- \( z + x=8 + 4 = 12\)

Step3: Calculate the perimeter

The perimeter \( P=(x + y)+(y + z)+(z + x)=10 + 14+12 = 44\) units.

Answer:

44 units