Using coordinate geometry, we can find the coordinates of a point which divides a given line segment in a given ratio. You have learnt that under the Midpoint and Section formulae. The midpoint of a line segment is the point on the segment that is equidistant from the endpoints. It can be represented as the average of its x coordinates and the average of its y-coordinates. For other ratios besides the 1 to 1 ratio, it is necessary to determine the total number of parts that the line segment must be divided into. For example, to determine the point that divides the segment AB into a 2:3 ratio, the segment must be divided into 2 + 3 = 5 equal parts.
In this lesson, you will learn how to do this using simple constructions.
To divide a line segment in the ratio m : n
- Draw any ray AX, making an acute angle with the line segment AB
- You will need to mark m + n equidistant points on the ray AX
i.e., for example:
if the points are A1, A2, A3, A4 and A5 on AX then AA1 = A1A2 = A2A3 = A3A4 = A4A5
- Join the line BAm+n. Draw a line through Am parallel to BAm+n. Name the point as C where the line intersects the line AB.
- Then, AC : CB = m : n
To construct a triangle similar to another triangle given the scale factor
- Construct span id="MathJax-Element-1-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="">ΔΔDEF similar to span id="MathJax-Element-2-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="">ΔΔABC with DE = span id="MathJax-Element-3-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="34">3434AB, EF = span id="MathJax-Element-4-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="34">3434BC and DF = span id="MathJax-Element-5-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="34">3434AC.
- Draw any ray BX making an acute angle with BC on the side opposite to the vertex A
- Locate 4 (the greater of 3 and 4 in span id="MathJax-Element-6-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="34">3434) points span id="MathJax-Element-7-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="B1B2,B3">B1B2,B3B1B2,B3and span id="MathJax-Element-8-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="B4">B4B4 on BX so that span id="MathJax-Element-9-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="BB1=B1B2=B2B3=B3B4">BB1=B1B2=B2B3=B3B4BB1=B1B2=B2B3=B3B4
- Join span id="MathJax-Element-10-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="B4">B4B4C and draw a line through span id="MathJax-Element-11-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="B3">B3B3 (the 3rd point, 3 being smaller of 3 and 4 in span id="MathJax-Element-12-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="34">3434) parallel to span id="MathJax-Element-13-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="B4">B4B4C to intersect BC at C′
- Draw a line through C′ parallel to the line CA to intersect BA at A′
- span id="MathJax-Element-14-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="">ΔΔA′BC′ is the required triangle.
2. Construction of Tangents to a Circle
Recall that the tangent to a circle is a line touching the circle at exactly one point. The point at which the tangent touches the circle is known as the ‘point of contact’.
You also know that the radius of the circle and tangent are perpendicular to each other at the point of contact and that there can be only one tangent at any point on the circumference of a circle.
In this lesson, you will learn how to use these to construct the tangent to a circle at a given point
Construction of a tangent to a circle with center O from a point P lying outside it
- Join PO and construct its perpendicular bisector. Let M be the midpoint of PO.
- Taking M as center and MO as radius, draw a circle. Let it intersect the given circle at the points Q and R.
- Join PQ and PR. Then PQ and PR are the required two tangents.
- If center of the circle is not given, you may locate its center first by taking any two non-parallel chords and then find the point of intersection of their perpendicular bisectors.