## Task 4 - 260724

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0494.jpg?height=117&width=1008&top_left_y=341&top_left_x=524)

Given two line segment lengths $P Q$ and $R S$ (see figure), there are two additional line segment lengths $a$ and $b$ that satisfy the conditions

(1) $P Q=2 a+b$,

(2) $R S=2 a-b$

and are uniquely determined by these conditions.

They are to be determined in two different ways:

(a) Transfer $P Q$ and $R S$ to a drawing sheet and construct (without using a length scale) the desired $a$ and $b$ from these given lengths! Describe your construction! Explain why your construction solves the problem (1) and (2)!

(b) Determine $a$ and $b$ computationally if the given lengths are $P Q=9.8 \mathrm{~cm}$ and $R S=6.6 \mathrm{~cm}$!
(a) I. Construction:

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0494.jpg?height=128&width=1014&top_left_y=1158&top_left_x=521)

II. Description:

1. Construct a segment \( U V \) of length \( U V = P Q \).
2. Extend \( U V \) beyond \( V \) by \( R S \) to point \( W \).
3. Construct the midpoint \( M \) of segment \( U W \).
4. Construct the midpoint \( K \) of segment \( U M \).

Then \( U K = a \) and \( M V = b \) are the desired lengths.

III. Justification:

For the constructed lengths, it follows that: From 4., \( U M = 2 \cdot U K = 2a \); from this and 1., it follows that \( P Q = U V = U M + M V = 2a + b \), i.e., (1) is satisfied.

From 3., \( M W = U M = 2a \); from this and 2., it follows that \( R S = V W = M W - M V = 2a - b \), i.e., (2) is satisfied.

(b) From (1) and (2), i.e., \( 2a + b = 9.8 \, \text{cm} \) (3) and \( 2a - b = 6.6 \, \text{cm} \) (4), it follows by addition that \( 4a = 16.4 \, \text{cm} \), thus \( a = 4.1 \, \text{cm} \). From this and (3), it follows that \( b = 9.8 \, \text{cm} - 8.2 \, \text{cm} = 1.6 \, \text{cm} \).

Solutions of the Second Round 1986 adopted from [5]

### 4.28.3 Third Round 1986, Class 7